Boxes

Martin has $n$ boxes labeled with positive integers from $1$ to $n$. Each box contains a toy. The toys are also labeled with positive integers from $1$ to $n$, and initially, the toy with label $i$ is contained in the box with label $i$.

From time to time, Martin calls one of his $m$ friends to come over and hang out. Once they meet up, his friend takes the toys out of the boxes and starts having fun with them. In the meantime, Martin is more interested in the boxes. Once they become bored, his friend puts the toys back into the boxes. However, he does not necessarily put every toy in the box it was taken from.

Martin has noticed that each of his $m$ friends scrambles the toys in the same way each time. More precisely, each of his friends has their own array of $n$ positive integers p_1, \ldots, p_n, which determines the way he will put the toys back into the boxes. Every positive integer from $1$ to $n$ appears exactly once in this array. His friend scrambles the toys such that, at the end of their meeting, the box with label $i$ contains the toy that was in the box with label p_i at the start of their meeting. Notice that, because every positive integer from $1$ to $n$ appears exactly once in the array, after all the toys are back in the boxes, each box will again have exactly one toy in it.

Martin is now interested in answering questions of the following form: he is wondering whether it is possible that the toy with label $a$ (which is initially in the box with label $a$) can end up in the box with label $b$ via a sequence of meetups with his friends. A sequence of meetups means that Martin can call whichever friends he wants and in any order. He can call a friend multiple times, or not at all. Martin is interested in answering $q$ such questions.