answer：Okay, I need to write a Python function to calculate the net population change of wild equines in a national park. The function will take two lists: births and deaths, each with 12 integers representing monthly counts. The output should be the total births minus total deaths over the year.Hmm, so first, I should think about how to approach this. Oh right, for each list, I need to sum all the elements. Then subtract the sum of deaths from the sum of births.Wait, what's the structure of the function? It should take two parameters, births and deaths. Then, compute the sum of each, subtract, and return the result.Let me think about the steps:1. Calculate the total births by summing all elements in the births list.2. Calculate the total deaths by summing all elements in the deaths list.3. Subtract total deaths from total births to get the net change.4. Return this net change.Oh right, and the function should handle cases where the net change is zero. But since subtraction naturally handles that, I don't need any special cases.So, how to implement this in Python.The function can be straightforward. Let's see:def calculate_population_change(births, deaths): total_births = sum(births) total_deaths = sum(deaths) net_change = total_births - total_deaths return net_changeWait, that's it? It seems simple enough. Let me test it with the example given.In the example, births sum is 10+12+15+9+8+13+14+11+10+13+12+9. Let's calculate that:10+12 is 22, +15 is 37, +9 is 46, +8 is 54, +13 is 67, +14 is 81, +11 is 92, +10 is 102, +13 is 115, +12 is 127, +9 is 136.Deaths sum is 8+7+9+6+5+7+8+6+7+5+6+4. Let's add them:8+7=15, +9=24, +6=30, +5=35, +7=42, +8=50, +6=56, +7=63, +5=68, +6=74, +4=78.So 136 -78 is 58? Wait, wait, no. Wait the example says the expected output is 62. Did I calculate wrong?Wait let me recheck the example:births = [10, 12, 15, 9, 8, 13, 14, 11, 10, 13, 12, 9]Let me add them step by step:10 +12 =22+15=37+9=46+8=54+13=67+14=81+11=92+10=102+13=115+12=127+9=136. That's correct.Deaths: [8,7,9,6,5,7,8,6,7,5,6,4]Adding:8+7=15+9=24+6=30+5=35+7=42+8=50+6=56+7=63+5=68+6=74+4=78.So 136-78 is 58, but the example says expected output is 62. Oh wait, did I make a mistake in the example?Wait the example says the expected output is 62. Hmm, perhaps I miscalculated the sums.Wait let me re-calculate the births:10, 12, 15, 9, 8, 13, 14, 11, 10, 13, 12, 9.Let's add them:10+12=2222+15=3737+9=4646+8=5454+13=6767+14=8181+11=9292+10=102102+13=115115+12=127127+9=136.Yes, that's correct.Deaths: 8+7=15, +9=24, +6=30, +5=35, +7=42, +8=50, +6=56, +7=63, +5=68, +6=74, +4=78.So 136-78 is 58, but the example expects 62. That's a problem. Did I read the example correctly?Wait looking back at the example:births = [10, 12, 15, 9, 8, 13, 14, 11, 10, 13, 12, 9]Wait wait, let me count the elements. There are 12 elements, correct.Wait maybe I made a mistake in adding the deaths.Deaths list is [8,7,9,6,5,7,8,6,7,5,6,4].Let me add them again:8 +7=1515+9=2424+6=3030+5=3535+7=4242+8=5050+6=5656+7=6363+5=6868+6=7474+4=78.Yes, that's correct.Hmm, but the example says the expected output is 62. So 136-78 is 58, but the example expects 62. So perhaps I made a mistake in the example's data.Wait let me recheck the example's data.In the example, the births list is [10, 12, 15, 9, 8, 13, 14, 11, 10, 13, 12, 9], which sums to 136.Deaths list is [8,7,9,6,5,7,8,6,7,5,6,4], which sums to 78.136-78 is 58, but the example expects 62. So perhaps the example is wrong, or I'm misunderstanding the problem.Wait wait, maybe I made a mistake in adding the births.Let me add the births again:10 +12 =22+15=37+9=46+8=54+13=67+14=81+11=92+10=102+13=115+12=127+9=136.Yes, that's correct.Wait, perhaps the example's expected output is wrong. Or perhaps I'm misunderstanding the problem.Wait, the problem says the function should return the net population change, which is total births minus total deaths. So in the example, 136-78=58, but the example expects 62. So that's a discrepancy.Wait, perhaps I made a mistake in the example's data. Let me check the example again.Wait, the example shows:births = [10, 12, 15, 9, 8, 13, 14, 11, 10, 13, 12, 9]deaths = [8,7,9,6,5,7,8,6,7,5,6,4]sum(births) is 136, sum(deaths) is 78. 136-78=58.But the expected output is 62. So perhaps the example is wrong, or I'm miscalculating.Alternatively, maybe the example has a typo. Let me see the sample code:In the example, the expected output is 62, which is sum(births) - sum(deaths). So perhaps the example's data is correct, but I'm miscalculating.Wait, perhaps I should re-calculate the sums again.Let me re-calculate the births:10, 12, 15, 9, 8, 13, 14, 11, 10, 13, 12, 9.Let's add them step by step:10 → 10+12 →22+15 →37+9 →46+8 →54+13 →67+14 →81+11 →92+10 →102+13 →115+12 →127+9 →136.Yes, that's correct.Deaths:8,7,9,6,5,7,8,6,7,5,6,4.Adding:8 →8+7 →15+9 →24+6 →30+5 →35+7 →42+8 →50+6 →56+7 →63+5 →68+6 →74+4 →78.So 136-78=58.But the example expects 62. So perhaps I'm misunderstanding the problem.Wait, maybe the function is supposed to return the absolute value? No, the problem says net change, which can be positive or negative.Alternatively, perhaps the example is wrong. Or perhaps I'm miscounting the elements.Wait, let me count the elements in the example's lists.Births list has 12 elements: 10,12,15,9,8,13,14,11,10,13,12,9 → yes, 12.Deaths list has 12 elements: 8,7,9,6,5,7,8,6,7,5,6,4 → yes, 12.So the sums are correct. So perhaps the example is wrong. Or perhaps I made a mistake in the problem statement.Wait, looking back at the problem statement, the example says the expected output is 62. But according to my calculations, it should be 58.Hmm, perhaps I should proceed with the function as described, regardless of the example. Because the function is correct as per the problem statement.So the function is correct. The example might have a typo.So the function is as I wrote before.So the code is:def calculate_population_change(births, deaths): total_births = sum(births) total_deaths = sum(deaths) return total_births - total_deathsTesting this with the example would give 58, but the example expects 62. So perhaps the example's data is incorrect.But perhaps I made a mistake in the example's data. Let me recheck.Wait, perhaps the example's data is correct, but I'm miscalculating.Wait, perhaps the births list is [10, 12, 15, 9, 8, 13, 14, 11, 10, 13, 12, 9], which is 12 elements.Let me add them again:10 +12 =2222+15=3737+9=4646+8=5454+13=6767+14=8181+11=9292+10=102102+13=115115+12=127127+9=136.Yes, that's correct.Deaths: 8+7=15, +9=24, +6=30, +5=35, +7=42, +8=50, +6=56, +7=63, +5=68, +6=74, +4=78.So 136-78=58.So the function as written is correct, but the example's expected output is wrong. Or perhaps the example's data is different.Wait, perhaps the example's data is different. Let me check the example again.Wait, in the example, the births list is [10, 12, 15, 9, 8, 13, 14, 11, 10, 13, 12, 9], which is 12 elements.Wait, perhaps I made a mistake in the deaths list. Let me recheck the deaths list.Deaths list: [8,7,9,6,5,7,8,6,7,5,6,4].Wait, perhaps I should add them again.Let me add each element:8 →87 →159 →246 →305 →357 →428 →506 →567 →635 →686 →744 →78.Yes, that's correct.Hmm, I'm confused. So perhaps the example is wrong, but the function is correct.So I'll proceed with writing the function as described.

answer：Okay, I need to write a Python function called longest_common_prefix that takes a list of DNA sequences and finds the longest common prefix among them. Hmm, let's think about how to approach this.First, what's a prefix? It's the starting part of each string. So the longest common prefix is the longest string that starts all the sequences in the list.So, how do I find this? Well, I think I can compare the characters of each string one by one, starting from the beginning. The moment a character doesn't match across all strings, I stop and return the prefix up to that point.Wait, but how do I handle varying lengths? Like, if one string is shorter than others, the prefix can't be longer than that string. So the maximum possible length is the length of the shortest string in the list.So, step by step:1. Check if the list is empty. If it is, return an empty string.2. Find the length of the shortest string in the list. That's the maximum possible prefix length.3. Iterate over each character position from 0 to max_length - 1.4. For each position, check if all strings have the same character at that position.5. If they do, add that character to the result.6. If not, break the loop and return the result so far.7. If all positions match up to the max_length, return the entire shortest string.Wait, but what if the list has only one string? Then the entire string is the prefix.Let me think about some examples.Example given: ["ATCGTAC", "ATCG", "ATCGA", "ATCGTAG"]The shortest string is "ATCG" with length 4. So we check each position up to 3 (0-based index).At position 0: all have 'A' → add to result.Position 1: all have 'T' → add.Position 2: all have 'C' → add.Position 3: all have 'G' → add.So the result is "ATCG".Another example: ["A", "A", "A"] → returns "A".If the list is ["A", "B"], then the first character doesn't match, so return empty string.What about empty strings in the list? According to the problem statement, each string is a DNA sequence, so probably they are non-empty. But the function should handle cases where the list is empty or has empty strings.Wait, the function's input is a list of strings, which can be empty. So if the list is empty, return empty. If any string is empty, then the common prefix is empty.So, in the code, first, handle the case where the list is empty: return "".Then, find the minimum length among the strings. If any string is empty, min_length is zero, so the loop won't run, and the result is empty.So, the steps in code:- If not dna_sequences: return ""- Get the minimum length of the sequences. Let's call it min_len.- Initialize the result as an empty string.- For i in range(min_len): - Get the i-th character of the first string as current_char. - For each sequence in dna_sequences: - If sequence[i] != current_char, return the result so far. - If all are same, add current_char to result.- After the loop, return the result.Wait, but what if the list has only one string? Then, the loop will run up to min_len, which is the length of that string, and the result will be the entire string.Yes, that's correct.Now, let's think about implementing this.In Python:def longest_common_prefix(dna_sequences): if not dna_sequences: return "" min_len = min(len(s) for s in dna_sequences) result = [] for i in range(min_len): current_char = dna_sequences[0][i] for s in dna_sequences: if s[i] != current_char: return ''.join(result) result.append(current_char) return ''.join(result)Wait, but what if all the sequences are the same? Like ["ATCG", "ATCG", "ATCG"], then the code will loop through all min_len (4) and return "ATCG".Yes.Another test case: ["ATC", "ATCG", "ATCGA"] → min_len is 3. So check each of the first 3 characters. All are same, so return "ATC".What about ["ATCG", "ATCC", "ATCA"] → the third character is 'G', 'C', 'A' in each. So at i=2, the current_char is 'G' from the first string. The second string has 'C' which is different. So the loop breaks, and returns "AT".Wait, no. Let's see:i=0: all 'A's → add to result.i=1: all 'T's → add.i=2: first string is 'C', second is 'C', third is 'C'? Wait, no, the first string is "ATCG" → index 2 is 'C'. Second string is "ATCC" → index 2 is 'C'. Third string is "ATCA" → index 2 is 'C'. So all same. Add 'C' → result is "ATC".i=3: first string is 'G', second is 'C', third is 'A'. So when checking the second string, s[3] is 'C' which is not equal to 'G'. So return "ATC".Yes.So the code seems to handle that.Another test case: ["", "ATCG"] → the min_len is 0. So the loop doesn't run, returns empty string.What about ["A", ""] → min_len is 0, returns empty.What about ["", ""] → returns empty.What about ["A"] → returns "A".So the code should handle all these cases.I think this should work.Wait, but what about the case where all sequences are empty? Like ["", "", ""] → min_len is 0, returns "".Yes.So, the code seems solid.

answer：Okay, I need to write a Python function that calculates the total practice time a child spends on music over a week. The input is a list of 7 integers, each representing the practice time for each day from Monday to Sunday. The function should return the sum of all these integers.Hmm, so the problem is straightforward. I think I can approach this by summing all the elements in the list. But wait, I should make sure that the list has exactly 7 elements. Oh, but the function's input is given as a list of 7 integers, so maybe I don't need to handle that case. But perhaps the function should still work correctly even if the list is longer or shorter? Or maybe the function assumes that the input is correct as per the problem statement.Well, the problem says the function takes a list of 7 integers, so I can safely assume that the input is correct. So my task is just to sum all the elements.So the function can be as simple as returning the sum of the list. Let me think about how to implement this.The function will be called something like total_practice_time, and it takes one parameter, which is the list of daily times.So the code would be something like:def total_practice_time(daily_times): return sum(daily_times)Wait, that's it? Because summing a list of integers in Python is straightforward with the built-in sum function. So for example, if the input is [30,45,60,50,40,35,25], summing them gives 30+45=75, +60=135, +50=185, +40=225, +35=260, +25=285. Which matches the sample output.Yes, that should work. So the function is really simple. I don't think there's any edge case I'm missing because the problem specifies that each integer is the practice time, and the list is exactly 7 days. So no need to handle empty lists or anything like that.I think that's all. So the code is just returning the sum of the list.

answer：Okay, I need to write a Python function to determine if a list of movies contains any titles performed by crossover artists. The function should take three inputs: a list of movie titles, a dictionary mapping each movie title to its cast, and a list of crossover artists. The output should be a list of movie titles that include at least one of these crossover artists.Let me think about how to approach this. So, the steps I need to follow are:1. Iterate through each movie title in the given list.2. For each movie, check if it exists in the movie casts dictionary. Because sometimes, maybe the movie isn't in the dictionary? Or wait, the function's first input is a list of movie titles, but the dictionary contains all the movies with their casts. So perhaps the first input is redundant, but maybe it's better to process only the movies in that list. Hmm, but the example shows that the first input is ["Film A", "Film B", "Film C"], and the dictionary has all of them. So I think the function should process each movie in the first list, regardless of the dictionary's keys. Wait, but what if a movie in the first list isn't present in the dictionary? Then, perhaps it has no actors, so it can't have any crossover artists. So for each movie in the first list, I need to check if it's in the dictionary. If it is, then look at its cast.Wait, no. The first input is the list of movie titles to consider. So for each title in that list, I should check if that title is present in the movie casts dictionary. Because if it's not, then there are no actors for that movie, so it won't have any crossover artists. So for each movie in the first list, I check if it's a key in the movie casts dict. If it is, then I look at its cast.3. For each movie in the first list, check if any of the actors in its cast are in the crossover artists list. If yes, add the movie title to the result list.So the plan is:- Initialize an empty list to hold the result.- Loop through each movie title in the given list of movie titles.- For each title, check if it exists in the movie casts dictionary.- If it does, get the list of actors for that movie.- Check if any of these actors are in the crossover artists list.- If at least one is found, add the movie title to the result list.How to implement this in Python.Let me think about the code structure.The function will be something like:def find_crossover_movies(movie_titles, movie_casts, crossover_artists): result = [] for title in movie_titles: if title in movie_casts: cast = movie_casts[title] for actor in cast: if actor in crossover_artists: result.append(title) break # No need to check other actors once one is found return resultWait, but what if a movie has multiple actors, some of whom are crossover artists. We only need to check if at least one is present, so once we find one, we can add the title and move on.Yes, that makes sense.Let me test this logic with the example provided.Example inputs:Movie titles: ["Film A", "Film B", "Film C"]Movie casts: {"Film A": ["Actor 1", "Model 1"], "Film B": ["Actor 2"], "Film C": ["Model 2", "Actor 3"]}Crossover artists: ["Model 1", "Model 2"]Processing each movie:- Film A: cast is ["Actor 1", "Model 1"]. Model 1 is in crossover, so add Film A.- Film B: cast is ["Actor 2"], not in crossover, so skip.- Film C: cast is ["Model 2", ...]. Model 2 is in crossover, so add Film C.Result is ["Film A", "Film C"], which matches the example.What about edge cases?Case 1: A movie in the titles list is not in the casts dict. For example, if the titles include "Film D" but it's not in the dict. Then, it's not added to the result.Case 2: A movie's cast is empty. Then, it can't have any crossover artists, so it's not added.Case 3: Multiple crossover artists in a movie's cast. But once one is found, the movie is added, and we break out of the loop for that movie.What about the efficiency? Well, for each movie, we loop through its cast until we find a match. This is efficient because as soon as a match is found, we stop checking the rest of the cast for that movie.Another point: the crossover artists list could be large. So checking 'actor in crossover_artists' might be slow if done with a list. To optimize, perhaps convert the crossover_artists list into a set for O(1) lookups. Because in Python, checking membership in a set is faster than in a list.So in the function, I can convert the crossover_artists list into a set at the beginning.So modifying the function:crossover_set = set(crossover_artists)Then, in the inner loop, check if actor is in crossover_set.This would make the membership check faster, especially if the list is large.So the updated function:def find_crossover_movies(movie_titles, movie_casts, crossover_artists): result = [] crossover_set = set(crossover_artists) for title in movie_titles: if title in movie_casts: cast = movie_casts[title] for actor in cast: if actor in crossover_set: result.append(title) break return resultYes, that should work.Testing this with the example:crossover_set is {"Model 1", "Model 2"}For Film A, "Model 1" is in the set, so added.Film C: "Model 2" is in the set, so added.Another test case: what if a movie has multiple crossover artists?E.g., cast is ["Model 1", "Model 2"], but once the first is found, it's added.What about a movie that has no crossover artists? It's not added.What about a movie that isn't in the casts dict? It's ignored.What about a movie in the titles list that's not in the casts dict? It's not added.Yes, that's correct.So the function should handle all these cases.I think this should solve the problem.

answer：Okay, so I have this problem about someone overcoming their fear of dogs through therapy. The time they spend with dogs each session is modeled by a polynomial function, ( T(n) = 4n^2 + 3n + 2 ), where ( n ) is the session number. There are two parts to this problem.First, I need to find the session number ( n ) where the time spent is exactly 50 minutes. That means I have to solve the equation ( 4n^2 + 3n + 2 = 50 ). Hmm, okay, let me write that down:( 4n^2 + 3n + 2 = 50 )I should subtract 50 from both sides to set the equation to zero:( 4n^2 + 3n + 2 - 50 = 0 )Simplifying that:( 4n^2 + 3n - 48 = 0 )Alright, now I have a quadratic equation. I can solve this using the quadratic formula. The quadratic formula is ( n = frac{-b pm sqrt{b^2 - 4ac}}{2a} ), where ( a = 4 ), ( b = 3 ), and ( c = -48 ).Let me compute the discriminant first: ( b^2 - 4ac = 3^2 - 4*4*(-48) ).Calculating that:( 9 - 4*4*(-48) = 9 + 16*48 )Wait, 4*4 is 16, and 16*48... let me compute that step by step. 16*40 is 640, and 16*8 is 128, so 640 + 128 = 768. So the discriminant is 9 + 768 = 777.So, discriminant is 777. That's a positive number, so we have two real solutions.Now, plugging back into the quadratic formula:( n = frac{-3 pm sqrt{777}}{2*4} )Simplify denominator: 2*4 is 8.So,( n = frac{-3 pm sqrt{777}}{8} )Now, let me compute ( sqrt{777} ). Hmm, 27 squared is 729, 28 squared is 784. So, ( sqrt{777} ) is between 27 and 28. Let me see, 27.5 squared is 756.25, which is less than 777. 27.8 squared is 772.84, still less. 27.9 squared is 778.41, which is just above 777. So, approximately, ( sqrt{777} ) is about 27.87.So, plugging that in:First solution:( n = frac{-3 + 27.87}{8} = frac{24.87}{8} approx 3.10875 )Second solution:( n = frac{-3 - 27.87}{8} = frac{-30.87}{8} approx -3.85875 )Since session numbers can't be negative, we discard the negative solution. So, ( n approx 3.10875 ). But since the session number has to be an integer, we need to check if session 3 or 4 is the one where the time is exactly 50 minutes.Wait, let me compute ( T(3) ) and ( T(4) ) to see.Compute ( T(3) = 4*(3)^2 + 3*3 + 2 = 4*9 + 9 + 2 = 36 + 9 + 2 = 47 ) minutes.Compute ( T(4) = 4*(4)^2 + 3*4 + 2 = 4*16 + 12 + 2 = 64 + 12 + 2 = 78 ) minutes.Hmm, so at session 3, it's 47 minutes, and at session 4, it's 78 minutes. But the equation gave me approximately 3.10875. So, that's between session 3 and 4. But since the person can't attend a fraction of a session, they would reach 50 minutes somewhere during session 4. But since the question is asking for the session number where they spend exactly 50 minutes, and the function is quadratic, it's not linear, so it's not necessarily that they reach exactly 50 minutes in a specific session. So, perhaps the answer is that they don't have an exact integer session where it's 50 minutes, but the closest is session 4, but that's 78 minutes. Wait, but 3.10875 is closer to 3, but session 3 is 47, which is less than 50, and session 4 is 78, which is way more.Wait, maybe I made a mistake in interpreting the problem. It says, "reach a session where they spend exactly 50 minutes." So, perhaps they need to find the smallest integer ( n ) such that ( T(n) geq 50 ). But the equation solution is approximately 3.10875, so the next integer is 4, which is session 4.But the problem says "exactly 50 minutes." So, perhaps the answer is that there is no integer session number where the time is exactly 50 minutes, but the first session where they spend more than 50 minutes is session 4.Wait, but the question is phrased as "reach a session where they spend exactly 50 minutes." So, maybe it's expecting the value of ( n ) even if it's not an integer? But in reality, the person can't attend a fraction of a session. So, perhaps the answer is that it's not possible, but the closest is session 4.Wait, but let me check the quadratic equation again. Maybe I made a calculation error.Wait, let me recompute the discriminant:( b^2 - 4ac = 3^2 - 4*4*(-48) = 9 + 768 = 777 ). That's correct.So, ( sqrt{777} ) is approximately 27.87, so ( n = (-3 + 27.87)/8 ≈ 24.87/8 ≈ 3.10875 ). So, that's correct.So, the exact solution is ( n = frac{-3 + sqrt{777}}{8} ), which is approximately 3.10875. So, since the person can't attend a fraction of a session, they would have to attend session 4 to spend more than 50 minutes. But the question is asking for the session number where they spend exactly 50 minutes. So, perhaps the answer is that there is no integer session number where the time is exactly 50 minutes, but the first session where they exceed 50 minutes is session 4.But the problem says "reach a session where they spend exactly 50 minutes." So, maybe the answer is that it's not possible, but the closest is session 4. However, the question is phrased as solving the equation, so perhaps it's expecting the exact value, even if it's not an integer.Wait, but in the context of the problem, the session number has to be an integer. So, perhaps the answer is that there is no integer solution, but the equation solution is approximately 3.11, so the person would need to attend session 4 to spend more than 50 minutes.But the question is specifically asking to solve the equation ( 4n^2 + 3n + 2 = 50 ). So, the mathematical solution is ( n = frac{-3 + sqrt{777}}{8} ), which is approximately 3.11. So, perhaps the answer is that the session number is approximately 3.11, but since sessions are integers, the person would need to attend session 4.But the problem might be expecting the exact value, so perhaps I should present both the exact solution and the approximate integer.Wait, let me check the problem again. It says, "determine the session number ( n ) by solving the equation." So, perhaps it's expecting the exact value, even if it's not an integer. So, the exact solution is ( n = frac{-3 + sqrt{777}}{8} ). But maybe I should rationalize or present it in a simplified form.Wait, 777 factors: 777 divided by 3 is 259, which is prime? Let me check: 259 divided by 7 is 37, because 7*37 is 259. So, 777 = 3*7*37. So, ( sqrt{777} ) doesn't simplify further. So, the exact solution is ( n = frac{-3 + sqrt{777}}{8} ).But in the context of the problem, since session numbers are integers, perhaps the answer is that the person would reach 50 minutes between session 3 and 4, but since they can't attend a fraction, they would need to attend session 4 to exceed 50 minutes.But the question is specifically asking to solve the equation, so perhaps the answer is the exact value, even if it's not an integer. So, I think I should present both the exact solution and note that it's approximately 3.11, but since sessions are integers, the next session is 4.Wait, but the problem doesn't specify whether ( n ) has to be an integer. It just says "session number ( n )", so perhaps ( n ) can be a real number, but in reality, it's an integer. So, perhaps the answer is that the session number is approximately 3.11, but since they can't have a fraction of a session, they would need to attend session 4.But the question is part 1, and part 2 is about the total time over the first ( k ) sessions. So, perhaps for part 1, it's expecting the exact solution, even if it's not an integer.So, to sum up, the solution to the equation is ( n = frac{-3 + sqrt{777}}{8} ), which is approximately 3.11. So, the person would reach 50 minutes around session 3.11, but since sessions are integers, they would need to attend session 4 to spend more than 50 minutes.But the problem says "exactly 50 minutes", so perhaps the answer is that there is no integer session number where the time is exactly 50 minutes, but the closest is session 4, which is 78 minutes.Wait, but let me check if I made a mistake in the quadratic equation. Maybe I should double-check my calculations.Wait, let me recompute ( T(3) ) and ( T(4) ):( T(3) = 4*(3)^2 + 3*3 + 2 = 4*9 + 9 + 2 = 36 + 9 + 2 = 47 ) minutes.( T(4) = 4*(4)^2 + 3*4 + 2 = 4*16 + 12 + 2 = 64 + 12 + 2 = 78 ) minutes.So, yes, 47 and 78. So, 50 is between 47 and 78, so the session number where they spend exactly 50 minutes is between 3 and 4, which is approximately 3.11.So, perhaps the answer is that the session number is approximately 3.11, but since they can't attend a fraction, they would need to attend session 4 to spend more than 50 minutes.But the problem is asking to "determine the session number ( n ) by solving the equation", so perhaps it's expecting the exact value, even if it's not an integer. So, the exact solution is ( n = frac{-3 + sqrt{777}}{8} ).Alternatively, maybe I made a mistake in setting up the equation. Let me check:The equation is ( 4n^2 + 3n + 2 = 50 ). Subtracting 50 gives ( 4n^2 + 3n - 48 = 0 ). That seems correct.So, quadratic formula: ( n = frac{-3 pm sqrt{9 + 768}}{8} = frac{-3 pm sqrt{777}}{8} ). Correct.So, the exact solution is ( n = frac{-3 + sqrt{777}}{8} ), approximately 3.11.So, for part 1, the answer is ( n = frac{-3 + sqrt{777}}{8} ), approximately 3.11.Now, moving on to part 2. The therapist suggests that the total time spent over the first ( k ) sessions should not exceed 150 minutes. So, we need to find the maximum ( k ) such that ( S(k) leq 150 ), where ( S(k) = sum_{n=1}^{k} T(n) ).Given ( T(n) = 4n^2 + 3n + 2 ), so ( S(k) = sum_{n=1}^{k} (4n^2 + 3n + 2) ).We can split this sum into three separate sums:( S(k) = 4sum_{n=1}^{k} n^2 + 3sum_{n=1}^{k} n + sum_{n=1}^{k} 2 ).We know the formulas for these sums:1. ( sum_{n=1}^{k} n^2 = frac{k(k+1)(2k+1)}{6} )2. ( sum_{n=1}^{k} n = frac{k(k+1)}{2} )3. ( sum_{n=1}^{k} 2 = 2k )So, substituting these into ( S(k) ):( S(k) = 4*frac{k(k+1)(2k+1)}{6} + 3*frac{k(k+1)}{2} + 2k )Simplify each term:First term: ( 4*frac{k(k+1)(2k+1)}{6} = frac{4}{6}k(k+1)(2k+1) = frac{2}{3}k(k+1)(2k+1) )Second term: ( 3*frac{k(k+1)}{2} = frac{3}{2}k(k+1) )Third term: ( 2k )So, combining all terms:( S(k) = frac{2}{3}k(k+1)(2k+1) + frac{3}{2}k(k+1) + 2k )Now, let's combine these terms. To do that, it's easier to have a common denominator. Let's use 6 as the common denominator.First term: ( frac{2}{3}k(k+1)(2k+1) = frac{4}{6}k(k+1)(2k+1) )Second term: ( frac{3}{2}k(k+1) = frac{9}{6}k(k+1) )Third term: ( 2k = frac{12}{6}k )So, now:( S(k) = frac{4k(k+1)(2k+1) + 9k(k+1) + 12k}{6} )Now, let's expand each term in the numerator:First term: ( 4k(k+1)(2k+1) )Let me expand ( (k+1)(2k+1) ) first:( (k+1)(2k+1) = 2k^2 + k + 2k + 1 = 2k^2 + 3k + 1 )So, first term: ( 4k*(2k^2 + 3k + 1) = 8k^3 + 12k^2 + 4k )Second term: ( 9k(k+1) = 9k^2 + 9k )Third term: ( 12k )So, combining all terms in the numerator:( 8k^3 + 12k^2 + 4k + 9k^2 + 9k + 12k )Combine like terms:- ( 8k^3 )- ( 12k^2 + 9k^2 = 21k^2 )- ( 4k + 9k + 12k = 25k )So, numerator is ( 8k^3 + 21k^2 + 25k )Thus, ( S(k) = frac{8k^3 + 21k^2 + 25k}{6} )So, the total time after ( k ) sessions is ( frac{8k^3 + 21k^2 + 25k}{6} ) minutes.We need to find the maximum integer ( k ) such that ( S(k) leq 150 ).So, set up the inequality:( frac{8k^3 + 21k^2 + 25k}{6} leq 150 )Multiply both sides by 6:( 8k^3 + 21k^2 + 25k leq 900 )So, we have:( 8k^3 + 21k^2 + 25k - 900 leq 0 )We need to solve for ( k ) in integers, so we can try plugging in integer values of ( k ) until the left side exceeds 900.Let me compute ( S(k) ) for k=1,2,3,... until it exceeds 150.But wait, let me compute ( S(k) ) for k=1:( S(1) = frac{8(1)^3 + 21(1)^2 + 25(1)}{6} = frac{8 + 21 + 25}{6} = frac{54}{6} = 9 ) minutes.k=2:( S(2) = frac{8(8) + 21(4) + 25(2)}{6} = frac{64 + 84 + 50}{6} = frac{198}{6} = 33 ) minutes.Wait, no, wait, that's not correct. Wait, ( 8k^3 ) for k=2 is 8*8=64, 21k^2=21*4=84, 25k=50. So, total numerator is 64+84+50=198. 198/6=33. Correct.k=3:( 8*27 + 21*9 + 25*3 = 216 + 189 + 75 = 480 ). 480/6=80.k=4:( 8*64 + 21*16 + 25*4 = 512 + 336 + 100 = 948 ). 948/6=158.Wait, 158 is greater than 150, so k=4 gives S(k)=158, which is over 150.So, let's check k=3: S(3)=80, which is under 150.Wait, but wait, let's compute S(4) correctly.Wait, 8k^3 for k=4 is 8*64=512.21k^2=21*16=336.25k=25*4=100.Total numerator: 512+336=848+100=948.948/6=158. Correct.So, S(4)=158>150.So, the maximum k where S(k) ≤150 is k=3, since S(3)=80 and S(4)=158.Wait, but wait, 80 is way below 150. Maybe I made a mistake in the calculations.Wait, let me compute S(k) for k=5:8*125=1000, 21*25=525, 25*5=125. Total numerator=1000+525=1525+125=1650. 1650/6=275.Wait, but that's way over. Wait, but k=4 is 158, which is over 150. So, k=3 is 80, which is under. So, maybe the maximum k is 3.But wait, that seems too low. Let me check my formula again.Wait, perhaps I made a mistake in the formula for S(k). Let me recompute S(k) for k=1,2,3,4 manually.Compute S(1)=T(1)=4*1 +3*1 +2=4+3+2=9. Correct.S(2)=T(1)+T(2)=9 + (4*4 +3*2 +2)=9 + (16+6+2)=9+24=33. Correct.S(3)=33 + T(3)=33 + (4*9 +3*3 +2)=33 + (36+9+2)=33+47=80. Correct.S(4)=80 + T(4)=80 + (4*16 +3*4 +2)=80 + (64+12+2)=80+78=158. Correct.So, S(4)=158>150, so the maximum k is 3.But wait, 158 is just slightly over 150, so maybe the person can attend 3 full sessions and part of the 4th session? But the problem says the total time over the first k sessions should not exceed 150. So, k must be an integer, so the maximum k is 3.But wait, let me check if k=3 is the maximum. Because S(3)=80, which is way below 150. Maybe I made a mistake in the formula for S(k). Let me recompute S(k) using the formula.Wait, the formula I derived was ( S(k) = frac{8k^3 + 21k^2 + 25k}{6} ). Let me compute S(3) using this formula:( (8*27 + 21*9 +25*3)/6 = (216 + 189 +75)/6 = (480)/6=80. Correct.S(4)= (8*64 +21*16 +25*4)/6=(512 +336 +100)/6=948/6=158. Correct.So, S(4)=158>150, so k=3 is the maximum.Wait, but 150 is much higher than 80, so maybe I made a mistake in the formula.Wait, let me compute S(k) manually for k=5:T(5)=4*25 +3*5 +2=100+15+2=117.S(5)=158 +117=275. Which is way over.Wait, but 150 is between S(3)=80 and S(4)=158. So, the maximum k where S(k) ≤150 is k=3, because S(4)=158>150.But wait, that seems counterintuitive because 150 is much higher than 80. Maybe I made a mistake in the formula.Wait, let me compute S(k) for k=4 again:T(1)=9, T(2)=24, T(3)=47, T(4)=78.Sum: 9+24=33, 33+47=80, 80+78=158. Correct.So, S(4)=158>150, so k=3 is the maximum.But wait, 150 is between S(3)=80 and S(4)=158. So, the person can attend 3 full sessions, which take 80 minutes total, and then part of the 4th session. But the problem says the total time over the first k sessions should not exceed 150. So, if k=4, the total is 158>150, which is over. So, the maximum k is 3.But wait, 80 is way below 150. Maybe the formula is wrong.Wait, let me recompute the sum formula.Given ( T(n) = 4n^2 + 3n + 2 ).So, ( S(k) = sum_{n=1}^{k} (4n^2 + 3n + 2) = 4sum n^2 + 3sum n + sum 2 ).Which is:( 4*frac{k(k+1)(2k+1)}{6} + 3*frac{k(k+1)}{2} + 2k ).Simplify:( frac{4k(k+1)(2k+1)}{6} + frac{3k(k+1)}{2} + 2k ).Simplify each term:First term: ( frac{4k(k+1)(2k+1)}{6} = frac{2k(k+1)(2k+1)}{3} ).Second term: ( frac{3k(k+1)}{2} ).Third term: ( 2k ).To combine, let's get a common denominator of 6.First term: ( frac{4k(k+1)(2k+1)}{6} ).Second term: ( frac{9k(k+1)}{6} ).Third term: ( frac{12k}{6} ).So, total:( frac{4k(k+1)(2k+1) + 9k(k+1) + 12k}{6} ).Expanding:First term: ( 4k(k+1)(2k+1) = 4k*(2k^2 + 3k +1) = 8k^3 +12k^2 +4k ).Second term: ( 9k(k+1) =9k^2 +9k ).Third term: (12k ).Adding all together:8k^3 +12k^2 +4k +9k^2 +9k +12k =8k^3 +21k^2 +25k.So, ( S(k) = frac{8k^3 +21k^2 +25k}{6} ). Correct.So, for k=3: (8*27 +21*9 +25*3)/6=(216+189+75)/6=480/6=80.k=4: (8*64 +21*16 +25*4)/6=(512+336+100)/6=948/6=158.So, correct.So, the maximum k where S(k) ≤150 is k=3, since S(4)=158>150.But wait, 150 is much higher than 80, so maybe the formula is wrong. Alternatively, perhaps I made a mistake in the sum.Wait, let me compute S(k) for k=5:T(5)=4*25 +3*5 +2=100+15+2=117.S(5)=158+117=275.Which is way over 150.Wait, but 150 is between S(3)=80 and S(4)=158. So, the maximum k is 3.But that seems odd because 150 is much higher than 80. Maybe the formula is correct, but the function T(n) increases rapidly because it's quadratic. So, the total time increases cubically, so it's possible that S(k) jumps from 80 to 158 when k increases from 3 to 4.So, perhaps the answer is k=3.But let me check if there's a way to have a non-integer k where S(k)=150, but since k must be an integer, the maximum k is 3.Alternatively, maybe I made a mistake in the formula for S(k). Let me try another approach.Alternatively, perhaps I should compute S(k) for k=4 and see if it's 158, which is over 150, so k=3 is the maximum.But let me check if I can solve the cubic equation ( 8k^3 +21k^2 +25k =900 ) for k.But since k must be an integer, and we've seen that k=3 gives 8k^3 +21k^2 +25k=480, which is less than 900, and k=4 gives 948, which is greater than 900, so the maximum k is 3.Wait, but 480 is much less than 900, so perhaps I made a mistake in the formula.Wait, no, because S(k)= (8k^3 +21k^2 +25k)/6. So, when S(k)=150, 8k^3 +21k^2 +25k=900.So, solving 8k^3 +21k^2 +25k -900=0.We can try to find integer roots. Let's try k=5: 8*125=1000, 21*25=525, 25*5=125. Total:1000+525+125=1650. 1650-900=750>0.k=4: 8*64=512, 21*16=336, 25*4=100. Total:512+336=848+100=948. 948-900=48>0.k=3: 8*27=216, 21*9=189, 25*3=75. Total:216+189=405+75=480. 480-900=-420<0.So, the equation crosses zero between k=3 and k=4. So, the real solution is between 3 and 4. But since k must be integer, the maximum k where S(k) ≤150 is k=3.But wait, S(3)=80, which is way below 150. So, perhaps the person can attend more sessions, but the total time would exceed 150 at k=4.But 80 is much less than 150, so maybe the formula is wrong.Wait, let me check the sum again.Wait, T(n)=4n²+3n+2.So, S(k)=sum_{n=1}^k (4n²+3n+2)=4*sum n² +3*sum n +sum 2.Sum n² from 1 to k is k(k+1)(2k+1)/6.Sum n is k(k+1)/2.Sum 2 is 2k.So, S(k)=4*(k(k+1)(2k+1)/6) +3*(k(k+1)/2) +2k.Simplify:= (4/6)k(k+1)(2k+1) + (3/2)k(k+1) +2k.= (2/3)k(k+1)(2k+1) + (3/2)k(k+1) +2k.To combine, let's get a common denominator of 6.= (4k(k+1)(2k+1) +9k(k+1) +12k)/6.Expanding:4k(k+1)(2k+1)=4k*(2k²+3k+1)=8k³+12k²+4k.9k(k+1)=9k²+9k.12k=12k.Total numerator:8k³+12k²+4k+9k²+9k+12k=8k³+21k²+25k.So, S(k)=(8k³+21k²+25k)/6.So, correct.So, for k=3, S(k)=80, k=4, S(k)=158.So, the maximum k is 3, because at k=4, it's over 150.But that seems odd because 150 is much higher than 80. Maybe the function T(n) is increasing too rapidly.Wait, let me compute T(n) for n=1 to 5:T(1)=4+3+2=9.T(2)=16+6+2=24.T(3)=36+9+2=47.T(4)=64+12+2=78.T(5)=100+15+2=117.So, the time per session is increasing rapidly. So, the total time S(k) is:k=1:9k=2:33k=3:80k=4:158k=5:275So, yes, it's increasing rapidly. So, the total time at k=4 is 158, which is over 150, so the maximum k is 3.So, the answer to part 2 is k=3.But wait, let me check if there's a way to have a non-integer k where S(k)=150, but since k must be an integer, the maximum k is 3.Alternatively, perhaps the person can attend 3 full sessions and part of the 4th session to reach exactly 150 minutes. But the problem says the total time over the first k sessions should not exceed 150. So, k must be an integer, so the maximum k is 3.So, summarizing:1. The session number where the time is exactly 50 minutes is ( n = frac{-3 + sqrt{777}}{8} approx 3.11 ), but since sessions are integers, the person would need to attend session 4 to spend more than 50 minutes.2. The maximum number of sessions ( k ) such that the total time does not exceed 150 minutes is 3.But wait, for part 1, the problem says "determine the session number ( n ) by solving the equation", so perhaps it's expecting the exact value, even if it's not an integer. So, the answer is ( n = frac{-3 + sqrt{777}}{8} ).But in the context of the problem, since session numbers are integers, perhaps the answer is that the person would reach 50 minutes between session 3 and 4, but since they can't attend a fraction, they would need to attend session 4.But the problem is part 1 and part 2, so perhaps for part 1, the answer is the exact value, and for part 2, the answer is k=3.So, final answers:1. ( n = frac{-3 + sqrt{777}}{8} ) or approximately 3.11.2. ( k = 3 ).But let me check if the problem expects integer solutions for both parts.For part 1, the problem says "determine the session number ( n )", so perhaps it's expecting an integer. But since the equation solution is not an integer, perhaps the answer is that there is no integer session number where the time is exactly 50 minutes, but the first session where the time exceeds 50 minutes is session 4.But the problem says "exactly 50 minutes", so perhaps the answer is that it's not possible, but the closest is session 4.Alternatively, perhaps the answer is the exact value, even if it's not an integer.Similarly, for part 2, the answer is k=3.So, to sum up:1. The session number ( n ) where the time is exactly 50 minutes is ( n = frac{-3 + sqrt{777}}{8} approx 3.11 ). Since sessions are integers, the person would need to attend session 4 to spend more than 50 minutes.2. The maximum number of sessions ( k ) such that the total time does not exceed 150 minutes is 3.But the problem might be expecting just the mathematical solutions, regardless of the integer context.So, for part 1, the exact solution is ( n = frac{-3 + sqrt{777}}{8} ), and for part 2, the maximum k is 3.So, I think that's the answer.

answer：Okay, I need to write a Python function that processes a list of animal observations and returns a dictionary with the total counts for each species. Hmm, let's think about how to approach this.So, each observation is a string like "species name - count". For example, "iguana - 3". My goal is to split each string into the species name and the count, then sum the counts for each species.First, I'll need to loop through each string in the input list. For each string, I'll split it into two parts: the species name and the count. How can I split the string? Oh right, using the split() method. But wait, the string is separated by " - ", so maybe I can split on that exact substring.Wait, for example, "iguana - 3" split on " - " would give me ["iguana", "3"]. That makes sense. So for each string, I can split it into two parts: the first part is the species, the second is the count.Once I have the species and count, I need to convert the count from a string to an integer. Then, I'll update a dictionary where the key is the species name, and the value is the sum of all counts for that species.So, I'll initialize an empty dictionary. Then, for each observation in the list:1. Split the string into species and count_str.2. Convert count_str to an integer.3. Check if the species is already a key in the dictionary. If yes, add the count to the existing value. If not, create a new entry with the species as the key and the count as the value.Let me think about the steps in code.Initialize the dictionary:species_counts = {}Then, loop through each observation in the input list:for observation in observations: # split the observation into species and count parts = observation.split(" - ") species = parts[0] count = int(parts[1]) # update the dictionary if species in species_counts: species_counts[species] += count else: species_counts[species] = countWait, that should work. But is there a more efficient way? Like using the get method. Alternatively, I can use a defaultdict from the collections module, which automatically initializes missing keys with a default value, like 0. But since the problem doesn't specify that I need to import anything, maybe it's better to stick with a regular dictionary.So, the code would look like that.Let me test this logic with the sample input.Sample input:["iguana - 3", "seagull - 5", "iguana - 2", "pelican - 4"]Processing each string:First string: "iguana - 3" → species is "iguana", count 3. Since it's not in the dict, add it with 3.Second: "seagull -5" → add with 5.Third: "iguana -2" → add 2 to the existing 3, total 5.Fourth: "pelican -4" → add 4.So the final dict is as expected.What about edge cases? Like an empty list? Well, the function should return an empty dict. What if a species appears multiple times? The code should handle that.Another test case: what if a species has a count of zero? According to the note, counts are non-negative, so zero is allowed. But adding zero won't change the sum, so it's fine.What about species names with hyphens? The note says species names do not contain hyphens, so we don't have to handle that. So the split on " - " is safe.So, the function can be written as:def process_observations(observations): species_counts = {} for obs in observations: parts = obs.split(" - ") species = parts[0] count = int(parts[1]) if species in species_counts: species_counts[species] += count else: species_counts[species] = count return species_countsWait, but what if the split results in more than two parts? For example, if a species name has a hyphen? Oh, but the note says species names do not contain hyphens, so that's not a problem. So each split will result in exactly two parts.So this code should handle all cases correctly.I think that's the solution.