Problem Statement

Given is a tree with $N$ vertices numbered $1$ to $N$, and $N-1$ edges numbered $1$ to $N-1$. Edge $i$ connects Vertex ai​ and bi​ bidirectionally and has a length of $1$.

Snuke will paint each vertex white or black. The niceness of a way of painting the graph is $\max (X,Y)$, where $X$ is the maximum among the distances between white vertices, and $Y$ is the maximum among the distances between black vertices. Here, if there is no vertex of one color, we consider the maximum among the distances between vertices of that color to be $0$.

There are 2N ways of painting the graph. Compute the sum of the nicenesses of all those ways, modulo (109+7).

Constraints

• 2≤N≤2×105
• 1≤ai​,bi​≤N
• The given graph is a tree.

Sample Input 1

2
1 2

Sample Output 1

2

• If we paint Vertex $1$ and $2$ the same color, the niceness will be $1$; if we paint them different colors, the niceness will be $0$.
• The sum of those nicenesses is $2$.

Sample Input 2

6
1 2
2 3
3 4
4 5
3 6

Sample Output 2

224

Sample Input 3

35
25 4
33 7
11 26
32 4
12 7
31 27
19 6
10 22
17 12
28 24
28 1
24 15
30 24
24 11
23 18
14 15
4 29
33 24
15 34
11 3
4 35
5 34
34 2
16 19
7 18
19 31
22 8
13 26
20 6
20 9
4 33
4 8
29 19
15 21

Sample Output 3

298219707

• Be sure to compute the sum modulo (109+7).