Three Paths on a Tree
Description
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.
Input
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.
Output
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.
Samples
8
1 2
2 3
3 4
4 5
4 6
3 7
3 8
5
1 8 6
Note
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.