Problem Description

As we know, Rikka is poor at math. Yuta is worrying about this situation, so he gives Rikka some math tasks to practice. There is one of them:

Now, Yuta has a tree with $n$ vertices. Let vertice 1 be the root and then for each vertice $i$ let $d_i$ be the distance between 1 and $i$.

Then Yuta wants to choose out at least two vertieces. It is clear that there have $2^n-n-1$ ways to choose. He will choose one of them equiprobable. Then let $f$ be the largest $d_i$ and $g$ be the second largest $d_i$ of the chosen vertices.($f$ may be equal to $g$)

Yuta wants to know the expected value of $\frac{(f+1)(g+1)}{f+1+g+1}$.

It is too difficult for Rikka. Can you help her?

Input

There are no more than 100 testcases and there are no more than 3 testcases with $n \geq 1000$

For each testcase, the first line contains a number $n(2 \leq n \leq 10^5)$. Then the second line contains $n-1$ numbers $f_i (1 \leq f_i \leq i)$, means the father of vertice $i+1$.

Output

For each testcase, print a single number. You only need to reserve six decimal places.

3
1 1

0.833333

Hint
There are four ways to choose vertices:
1.choose {1,2}, then f=1,g=0.
2.choose {1,3}, then f=1,g=0.
3.choose {2,3}, then f=1,g=1.
4.choose {1,2,3}, then f=1,g=1.