Christmas Tree

Takahashi has decided to make a Christmas Tree for the Christmas party in AtCoder, Inc.

A Christmas Tree is a tree with N vertices numbered 1 through N and N-1 edges, whose i-th edge (1\leq i\leq N-1) connects Vertex a_i and b_i.

He would like to make one as follows:

• Specify two non-negative integers A and B.
• Prepare A Christmas Paths whose lengths are at most B. Here, a Christmas Path of length X is a graph with X+1 vertices and X edges such that, if we properly number the vertices 1 through X+1, the i-th edge (1\leq i\leq X) will connect Vertex i and i+1.
• Repeat the following operation until he has one connected tree:
• Select two vertices x and y that belong to different connected components. Combine x and y into one vertex. More precisely, for each edge (p,y) incident to the vertex y, add the edge (p,x). Then, delete the vertex y and all the edges incident to y.
• Properly number the vertices in the tree.

Takahashi would like to find the lexicographically smallest pair (A,B) such that he can make a Christmas Tree, that is, find the smallest A, and find the smallest B under the condition that A is minimized.

Solve this problem for him.

Constraints

• 2 \leq N \leq 10^5
• 1 \leq a_i,b_i \leq N
• The given graph is a tree.

Input

Input is given from Standard Input in the following format:

N a_1 b_1 : a_{N-1} b_{N-1}

Output

For the lexicographically smallest (A,B), print A and B with a space in between.

Examples

Input

7 1 2 2 3 2 4 4 5 4 6 6 7

Output

3 2

Input

8 1 2 2 3 3 4 4 5 5 6 5 7 5 8

Output

2 5

Input

10 1 2 2 3 3 4 2 5 6 5 6 7 7 8 5 9 10 5

Output

3 4

inputFormat

Input

Input is given from Standard Input in the following format:

N a_1 b_1 : a_{N-1} b_{N-1}

outputFormat

Output

For the lexicographically smallest (A,B), print A and B with a space in between.

Examples

Input

7 1 2 2 3 2 4 4 5 4 6 6 7

Output

3 2

Input

8 1 2 2 3 3 4 4 5 5 6 5 7 5 8

Output

2 5

Input

10 1 2 2 3 3 4 2 5 6 5 6 7 7 8 5 9 10 5

Output

3 4

样例

7
1 2
2 3
2 4
4 5
4 6
6 7

3 2