"""# ### 谜题描述 Ashishgup and FastestFinger play a game. They start with a number n and play in turns. In each turn, a player can make any one of the following moves: * Divide n by any of its odd divisors greater than 1. * Subtract 1 from n if n is greater than 1. Divisors of a number include the number itself. The player who is unable to make a move loses the game. Ashishgup moves first. Determine the winner of the game if both of them play optimally. Input The first line contains a single integer t (1 ≤ t ≤ 100) — the number of test cases. The description of the test cases follows. The only line of each test case contains a single integer — n (1 ≤ n ≤ 10^9). Output For each test case, print \"Ashishgup\" if he wins, and \"FastestFinger\" otherwise (without quotes). Example Input 7 1 2 3 4 5 6 12 Output FastestFinger Ashishgup Ashishgup FastestFinger Ashishgup FastestFinger Ashishgup Note In the first test case, n = 1, Ashishgup cannot make a move. He loses. In the second test case, n = 2, Ashishgup subtracts 1 on the first move. Now n = 1, FastestFinger cannot make a move, so he loses. In the third test case, n = 3, Ashishgup divides by 3 on the first move. Now n = 1, FastestFinger cannot make a move, so he loses. In the last test case, n = 12, Ashishgup divides it by 3. Now n = 4, FastestFinger is forced to subtract 1, and Ashishgup gets 3, so he wins by dividing it by 3. Here is a reference code to solve this task. You can use this to help you genereate cases or validate the solution. ```python import math test_cases=int(input()) res=[] for i in range(test_cases): n=int(input()) t=0 while n%2==0: n=n//2 t+=1 if t==0: if n==1: res.append(\"FastestFinger\") else: res.append(\"Ashishgup\") elif t==1: if n==1: res.append(\"Ashishgup\") else: cp=0 for j in range(3,int(math.floor(math.sqrt(n)))+1,2): #print(j) if n%j ==0: cp=1 break if cp==1: res.append(\"Ashishgup\") else: res.append(\"FastestFinger\") else: if n==1: res.append(\"FastestFinger\") else: res.append(\"Ashishgup\") '''cp=0 for j in range(3,n//2,2): if n%j ==0: cp=1 break if cp==1: res.append(\"FastestFinger\") else: res.append(\"Ashishgup\")''' for i in range(test_cases): print(res[i]) ``` 请完成上述谜题的训练场环境类实现，包括所有必要的方法。 """ from bootcamp import Basebootcamp import re import math import random from bootcamp import Basebootcamp class Cnumbergamebootcamp(Basebootcamp): def __init__(self, **params): self.min_n = max(1, params.get('min_n', 1)) self.max_n = params.get('max_n', 10**9) self.params = params def case_generator(self): def generate_valid_n(): for _ in range(1000): # 增加尝试次数 case_type = random.choice([ 'edge_1', 'edge_2', 'prime', 'power_of_2', 'two_times_prime', 'two_times_composite', 'complex_case' ]) if case_type == 'edge_1': n = 1 elif case_type == 'edge_2': n = 2 elif case_type == 'prime': n = self._generate_odd_prime() elif case_type == 'power_of_2': max_exp = math.floor(math.log2(self.max_n)) if max_exp < 2: continue # 无法生成合理2^n exp = random.randint(2, max_exp) n = 2 ** exp elif case_type == 'two_times_prime': max_p = self.max_n // 2 min_p = max(3, self.min_n // 2) if min_p > max_p: continue p = self._generate_odd_prime(min_p, max_p) n = 2 * p elif case_type == 'two_times_composite': max_c = self.max_n // 2 if max_c < 9: # 最小奇合数9 continue c = self._generate_odd_composite(3, max_c) n = 2 * c elif case_type == 'complex_case': max_t = math.floor(math.log2(self.max_n)) if max_t < 2: continue t = random.randint(2, min(5, max_t)) base = 2 ** t remaining = self.max_n // base if remaining < 3: continue factors = self._generate_odd_composite(3, remaining) n = base * factors if self.min_n <= n <= self.max_n: return n return random.randint(self.min_n, self.max_n) n = generate_valid_n() return { 'n': n, 'correct_answer': self.get_correct_answer(n) } def _generate_odd_prime(self, min_p=3, max_p=None): max_p = max_p or self.max_n // 2 if min_p % 2 == 0: min_p += 1 attempts = 0 while attempts < 1000: p = random.randint(min_p, max_p) if p % 2 == 0: continue if self.is_prime(p): return p attempts += 1 return 3 # fallback def _generate_odd_composite(self, min_val=9, max_val=None): max_val = max_val or self.max_n // 2 while True: num = random.randint(min_val, max_val) if num % 2 == 0: continue if not self.is_prime(num): return num @staticmethod def is_prime(num): if num < 2: return False if num % 2 == 0: return num == 2 for i in range(3, int(math.isqrt(num)) + 1, 2): if num % i == 0: return False return True @staticmethod def get_correct_answer(n): original_n = n t = 0 while n % 2 == 0: n = n // 2 t += 1 k = n if t == 0: return "FastestFinger" if k == 1 else "Ashishgup" elif t == 1: if k == 1: return "Ashishgup" is_prime = Cnumbergamebootcamp.is_prime(k) return "FastestFinger" if is_prime else "Ashishgup" else: return "FastestFinger" if k == 1 else "Ashishgup" @staticmethod def prompt_func(question_case): n = question_case['n'] return f"""Ashishgup和FastestFinger在玩一个数字游戏。规则如下： - 初始数字为{n}。玩家轮流进行操作，Ashishgup先手。 - 每个回合可以选择以下操作之一： a) 将当前数除以一个大于1的奇数因子 b) 当当前数>1时，减去1 无法操作的玩家失败。两人都采用最优策略。 请分析游戏过程并输出获胜者姓名，将答案置于[answer]标签内。例如： [answer]Ashishgup[/answer]""" @staticmethod def extract_output(output): matches = re.findall(r'\[answer\]\s*(.*?)\s*\[/answer\]', output, re.IGNORECASE) if not matches: return None last_answer = matches[-1].strip().capitalize() return last_answer if last_answer in ['Ashishgup', 'FastestFinger'] else None @classmethod def _verify_correction(cls, solution, identity): return solution == identity.get('correct_answer')