Distance Query Cases

Case $1$: Both in tail. Check if $y$ is reachable from $x$ by jumping. If not, no path exists. If yes, compute distance by simulation or binary search with binary lifting. Case $2$: $x$ in tail, $y$ in cycle. Jump from $x$ to cycle, then compute steps around cycle to $y$.

Use cycle positions. Case $3$: Both in cycle. Compute steps around the cycle from $x$ to $y$ using $(\text{pos}[y] - \text{pos}[x]) \mod \lambda$. Case $4$: $x$ in cycle, $y$ in tail. No path exists because you cannot leave a cycle.