You are given a tree with $$$n$$$ vertices labeled $$$1, 2, \ldots, n$$$. The length of a simple path in the tree is the number of vertices in it.

You are to select a set of simple paths of length at least $$$2$$$ each, but you cannot simultaneously select two distinct paths contained one in another. Find the largest possible size of such a set.

Formally, a set $$$S$$$ of vertices is called a route if it contains at least two vertices and coincides with the set of vertices of a simple path in the tree. A collection of distinct routes is called a timetable. A route $$$S$$$ in a timetable $$$T$$$ is called redundant if there is a different route $$$S' \in T$$$ such that $$$S \subset S'$$$. A timetable is called efficient if it contains no redundant routes. Find the largest possible number of routes in an efficient timetable.

Input

The first line contains a single integer $$$n$$$ ($$$2 \le n \le 3000$$$).

The $$$i$$$-th of the following $$$n - 1$$$ lines contains two integers $$$u_i$$$ and $$$v_i$$$ ($$$1 \le u_i, v_i \le n$$$, $$$u_i \neq v_i$$$) — the numbers of vertices connected by the $$$i$$$-th edge.

It is guaranteed that the given edges form a tree.

Output

Print a single integer — the answer to the problem.

Note

In the first example, possible efficient timetables are $$$\{\{1, 2\}, \{1, 3\}, \{1, 4\}\}$$$ and $$$\{\{1, 2, 3\}, \{1, 2, 4\}, \{1, 3, 4\}\}$$$.

In the second example, we can choose $$$\{ \{1, 2, 3\}, \{2, 3, 4\}, \{3, 4, 6\}, \{2, 3, 5\}, \{3, 4, 5\}, \{3, 4, 7\}, \{4, 6, 7\}\}$$$.

Redundant Routes

Examples