Mock Euclid 2025 - Problem 8 Part B

The number of ways to choose four perfect squares and one cube such that their product equals 645^{245} can be expressed in the form p_1^{q_1} \cdot p_2^{q_2} \cdot p_3^{q_3} \cdots p_n^{q_n} where the p_i are distinct prime numbers and the q_i are positive integer exponents. Compute the sum p_1 + p_2 + \cdots + p_n + q_1 + q_2 + \cdots + q_n.