Holiday Contest 2025 - Team Round - Problem 9

Let f(n) return the number of positive divisors of n (including 1 and n), where n is a positive integer. The integers f(113), f(x), f(y), f(z), f(176) form an arithmetic sequence, in that order, where 113 \leq x \leq y \leq z \leq 176. What is the minimum value of x+y+z?