Where Is the Root?

This is an interactive problem.

You are given a tree of n vertices. The tree is a graph such that there is exactly one simple path between every pair of vertices. It's also guaranteed that at least one vertex is directly connected by an edge to at least $3$ vertices. One of the vertices is the root, and your task is to find it. In order to do this, you are allowed to ask queries of the following form:

• For a given set $a\_1 , a\_2 , \dots , a\_m$ of vertices, check if their lowest common ancestor is in this set.

A vertex $v$ is a common ancestor of a set $S$ of vertices if the paths from all vertices in $S$ to the root pass through $v$. The lowest common ancestor (LCA) of a set $S$ of vertices is the common ancestor of $S$ which is farthest from the root.

The hidden root is vertex $4$.

In the first query, the LCA of vertices $5$ and $6$ is vertex $3$ which is not among vertices $5$ and $6$ so the answer is "NO".

In the second query, the LCA of vertices $3$, $5$, and $6$ is vertex $3$ so the answer is "YES".

In the third query, the LCA of vertices $1$ and $7$ is vertex $4$ so the answer is "NO".

In the fourth query, the LCA of vertices $4$ and $6$ is vertex $4$ so the answer is "YES".

After that, we can guess that root is vertex $4$ which is the correct answer.