Counting Is Fun

Suppose $f(X, Y, Z)$ denotes the number of arrays $A$ of length $X$ such that: - $1 \leq A_i \leq Y$, for all $(1 \leq i \leq X)$; - There are **at most** $Z$ values of $i$ $(2 \leq i \leq X)$ such that $A_i=A_{i-1}$. In other words, for at most $Z$ indices, the element at a given index is equal to the previous element. You are given an integer $M$ and $Q$ queries. The $i^{th}$ query is of the f