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internbootcamp/bootcamp/cprefixproductsequence/cprefixproductsequence.py

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+"""# 
+
+### 谜题描述
+Consider a sequence [a1, a2, ... , an]. Define its prefix product sequence <image>.
+
+Now given n, find a permutation of [1, 2, ..., n], such that its prefix product sequence is a permutation of [0, 1, ..., n - 1].
+
+Input
+
+The only input line contains an integer n (1 ≤ n ≤ 105).
+
+Output
+
+In the first output line, print \"YES\" if such sequence exists, or print \"NO\" if no such sequence exists.
+
+If any solution exists, you should output n more lines. i-th line contains only an integer ai. The elements of the sequence should be different positive integers no larger than n.
+
+If there are multiple solutions, you are allowed to print any of them.
+
+Examples
+
+Input
+
+7
+
+
+Output
+
+YES
+1
+4
+3
+6
+5
+2
+7
+
+
+Input
+
+6
+
+
+Output
+
+NO
+
+Note
+
+For the second sample, there are no valid sequences.
+
+Here is a reference code to solve this task. You can use this to help you genereate cases or validate the solution.
+```python
+from math import sqrt
+n=input()
+if n==4 :
+    print \"YES\"
+    print 1
+    print 3
+    print 2
+    print 4
+elif n==1:
+    print \"YES\"
+    print 1
+else :
+    prime=[1]*(n+1)
+    prime[0]=0
+    prime[1]=0
+    i=2
+    while i<= int(sqrt(n))+1:
+        if prime[i]==1:
+            for s in range(i*i,n+1,i):
+                prime[s]=0
+        if prime[n]==0:
+            break
+        i=i+1
+    if prime[n]:
+        print \"YES\"
+        print 1
+        i=2
+        while i<n:
+            print ((i*pow(i-1,n-2,n))%n)
+            i=i+1
+        print n
+    else:
+        print \"NO\"
+```
+
+
+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 random
+from math import isqrt
+from bootcamp import Basebootcamp
+
+class Cprefixproductsequencebootcamp(Basebootcamp):
+    def __init__(self, max_n=1e5, **kwargs):
+        super().__init__(**kwargs)
+        self.max_n = int(max_n)
+    
+    def sieve(self):
+        """Optimized sieve with bytearray storage"""
+        if self.max_n < 2:
+            return []
+        sieve = bytearray([1])*(self.max_n+1)
+        sieve[0] = sieve[1] = 0
+        for i in range(2, isqrt(self.max_n)+1):
+            if sieve[i]:
+                sieve[i*i : self.max_n+1 : i] = b'\x00'*len(sieve[i*i : self.max_n+1 : i])
+        return [i for i, v in enumerate(sieve) if v]
+
+    def case_generator(self):
+        # Valid case candidates
+        valid = {1, 4}
+        primes = set(self.sieve())
+        valid.update(primes)
+        
+        # Generate case type
+        candidates = list(range(1, self.max_n+1))
+        exist_cases = [n for n in candidates if n in valid]
+        non_exist_cases = [n for n in candidates if n not in valid]
+        
+        # Ensure balanced case generation
+        if exist_cases and (random.random() < 0.5 or not non_exist_cases):
+            n = random.choice(exist_cases)
+            return {'n': n, 'exists': True}
+        elif non_exist_cases:
+            return {'n': random.choice(non_exist_cases), 'exists': False}
+        else:  # Fallback when all cases are valid
+            return {'n': 1, 'exists': True}
+    
+    @staticmethod
+    def prompt_func(case):
+        n = case['n']
+        return f"""Given n={n}, determine if there exists a permutation of 1-{n} where: 
+1. All prefix products modulo {n} 
+2. Form a permutation of 0-{n-1}
+
+Output format (inside [answer] tags):
+[answer]
+YES
+<p1>
+<p2>
+...
+<p{n}>
+[/answer]
+OR
+[answer]
+NO
+[/answer]"""
+
+    @staticmethod
+    def extract_output(text):
+        # Robust extraction with negative lookahead
+        pattern = r'\[answer\][\s]*((?!\[answer\]).*?)[\s]*\[/answer\]'
+        matches = re.findall(pattern, text, re.DOTALL|re.IGNORECASE)
+        if not matches:
+            return None
+        
+        content = matches[-1].strip().upper()
+        lines = [l.strip() for l in content.split('\n')]
+        
+        if lines[0].startswith('NO'):
+            return {'answer': 'NO'} if len(lines) == 1 else None
+        
+        if lines[0].startswith('YES') and len(lines) == int(lines[0][3:].strip() or 0)+1:
+            try:
+                nums = list(map(int, lines[1:]))
+                return {'answer': 'YES', 'sequence': nums}
+            except ValueError:
+                pass
+        return None
+
+    @classmethod
+    def _verify_correction(cls, solution, case):
+        # Structural validation
+        if not solution or case['exists'] != (solution.get('answer') == 'YES'):
+            return False
+        
+        if solution['answer'] == 'NO':
+            return True
+        
+        # Numerical validation
+        n = case['n']
+        seq = solution.get('sequence', [])
+        if sorted(seq) != list(range(1, n+1)):
+            return False
+        
+        # Prefix product verification
+        seen = set()
+        product = 1
+        for num in seq:
+            product = (product * num) % n
+            if product in seen:
+                return False
+            seen.add(product)
+        return seen == set(range(n))