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. 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. Print a single integer — the answer to the problem. 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\}\}).