CF1294F.Three Paths on a Tree

You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.

Your task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$ , $b$ and $c$ , or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.

The simple path is the path that visits each vertex at most once.

The first line contains one integer number $n$ ( $3 \le n \le 2 \cdot 10^5$ ) — the number of vertices in the tree.

Next $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ( $1 \le a_i$ , $b_i \le n$ , $a_i \ne b_i$ ). It is guaranteed that given graph is a tree.

In the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$ , $b$ and $c$ , or $a$ and $c$ .

In the second line print three integers $a, b, c$ such that $1 \le a, b, c \le n$ and $a \ne, b \ne c, a \ne c$ .

If there are several answers, you can print any.

The picture corresponding to the first example (and another one correct answer):

If you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$ , the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$ . The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$ . It can be shown that there is no better answer.