Permutation Cycle

You are given a permutation $P$ of the integers $1$ to $N$, and an integer $K$. For this permutation $P$, let $G_P$ denote the undirected graph on $N$ vertices, which has edges between $i$ and $P_i$ for each $1 \leq i \leq N$ (and no other edges). It is well-known that each connected component of $G_P$ is a cycle. For each $x = 1, 2, \ldots, N$, solve the following problem **independently**: