To solve this problem, we need to determine the number of distinct prime numbers that can be formed using the digits of a given input number. The solution involves generating all possible permutations of the digits (of any length) and checking each permutation for primality, while ensuring we count only unique primes.

Approach

1. Prime Check Function: First, we need a helper function to check if a number is prime. This function efficiently checks for primality by testing divisibility up to the square root of the number.
2. Generate Permutations: For each possible length from 1 to the length of the input number, generate all permutations of the digits.
3. Filter Valid Numbers: Skip permutations with leading zeros (except for single-digit zero, which is not prime).
4. Collect Unique Primes: Use a set to collect all valid primes to avoid duplicates.
5. Count Primes: The size of the set gives the number of distinct primes.

Solution Code

import itertools

def is_prime(n):
    if n <= 1:
        return False
    if n == 2:
        return True
    if n % 2 == 0:
        return False
    for i in range(3, int(n ** 0.5) + 1, 2):
        if n % i == 0:
            return False
    return True

def count_distinct_primes(s):
    primes = set()
    length = len(s)
    for l in range(1, length + 1):
        for perm in itertools.permutations(s, l):
            num_str = ''.join(perm)
            if len(num_str) > 1 and num_str[0] == '0':
                continue
            num = int(num_str)
            if is_prime(num):
                primes.add(num)
    return len(primes)

# Read input and output the result
input_str = input().strip()
print(count_distinct_primes(input_str))

Explanation

1. Prime Check: The is_prime function checks if a number is prime by first handling small cases (0,1,2) and then testing divisibility by odd numbers up to the square root of the number. This ensures efficiency.
2. Permutations: Using itertools.permutations, we generate all possible sequences of digits of varying lengths. This covers all possible numbers that can be formed from the input digits.
3. Leading Zeros: We skip permutations that start with zero (for lengths greater than 1) since these are not valid numbers (e.g., "03" is not a valid 2-digit number).
4. Unique Primes: Using a set to store primes ensures that even if different permutations result in the same prime number, it is counted only once.
5. Result: The size of the set of primes gives the count of distinct primes that can be formed from the input digits.

This approach efficiently handles all edge cases (like duplicate digits, leading zeros, and small numbers) and provides the correct count of distinct primes. The use of sets ensures we avoid duplicates, and the prime check is optimized for performance.

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