Permutation Composition Queries

You are given $N$ sequences $P_1, P_2, \dots, P_N$. Each of them is a permutation of numbers $1$ through $M$. Let's define a product of two permutations $X$ and $Y$ of numbers $1$ through $M$ as a permutation $Z = X\cdot Y$ such that $Z_i=Y_{X_i}$ for each valid $i$. You should answer $Q$ queries. Each query is described by two numbers $L$ and $R$. Let's define a permutation $B = P_L \cdot P_{L+