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F. Cowmpany Cowmpensation

time limit per test

2 secondsmemory limit per test

256 megabytesinput

standard inputoutput

standard outputAllen, having graduated from the MOO Institute of Techcowlogy (MIT), has started a startup! Allen is the president of his startup. He also hires $$$n-1$$$ other employees, each of which is assigned a direct superior. If $$$u$$$ is a superior of $$$v$$$ and $$$v$$$ is a superior of $$$w$$$ then also $$$u$$$ is a superior of $$$w$$$. Additionally, there are no $$$u$$$ and $$$v$$$ such that $$$u$$$ is the superior of $$$v$$$ and $$$v$$$ is the superior of $$$u$$$. Allen himself has no superior. Allen is employee number $$$1$$$, and the others are employee numbers $$$2$$$ through $$$n$$$.

Finally, Allen must assign salaries to each employee in the company including himself. Due to budget constraints, each employee's salary is an integer between $$$1$$$ and $$$D$$$. Additionally, no employee can make strictly more than his superior.

Help Allen find the number of ways to assign salaries. As this number may be large, output it modulo $$$10^9 + 7$$$.

Input

The first line of the input contains two integers $$$n$$$ and $$$D$$$ ($$$1 \le n \le 3000$$$, $$$1 \le D \le 10^9$$$).

The remaining $$$n-1$$$ lines each contain a single positive integer, where the $$$i$$$-th line contains the integer $$$p_i$$$ ($$$1 \le p_i \le i$$$). $$$p_i$$$ denotes the direct superior of employee $$$i+1$$$.

Output

Output a single integer: the number of ways to assign salaries modulo $$$10^9 + 7$$$.

Examples

Input

3 2

1

1

Output

5

Input

3 3

1

2

Output

10

Input

2 5

1

Output

15

Note

In the first sample case, employee 2 and 3 report directly to Allen. The three salaries, in order, can be $$$(1,1,1)$$$, $$$(2,1,1)$$$, $$$(2,1,2)$$$, $$$(2,2,1)$$$ or $$$(2,2,2)$$$.

In the second sample case, employee 2 reports to Allen and employee 3 reports to employee 2. In order, the possible salaries are $$$(1,1,1)$$$, $$$(2,1,1)$$$, $$$(2,2,1)$$$, $$$(2,2,2)$$$, $$$(3,1,1)$$$, $$$(3,2,1)$$$, $$$(3,2,2)$$$, $$$(3,3,1)$$$, $$$(3,3,2)$$$, $$$(3,3,3)$$$.

Codeforces (c) Copyright 2010-2023 Mike Mirzayanov

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