\text{Given a set } A={a_1,a_2,\ldots,a_k} \text{ consisting of distinct real numbers, define }S(A) \text{ as the sum of all elements of } A:]S(A)=a_1+a_2+\cdots+a_k.$$$$\text{For example, if } p \text{ is a prime number and } A=\{1,p\}, \text{ the set of divisors of } p,\text{ then } S(A)=p+1.$$$$\text{Now, let } A \text{ be the set obtained by taking } \log_{10} \text{ of each divisor of }10^{2025}, $$$$\text{and multiplying the value by } +1 \text{ if the divisor is even, and by }-1 \text{ if the divisor is odd.}$$$$\text{Compute the product of the prime factors appearing in the prime factorization of }S(A)-2025\cdot 2026 \cdot \log_{10} 2.
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[THOUGHT] We need to compute the product of the distinct prime factors of the number 2025 * 2026 * 1012, which we found to be 2^3 * 3^4 * 5^2 * 11 * 23 * 1013. The distinct prime factors are 2, 3, 5, 11, 23, and 1013. We will use Python to calculate the product of these primes. [/THOUGHT] [PYTHON] print(2 * 3 * 5 * 11 * 23 * 1013) [/PYTHON]
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