CSES - Company Queries II
  • Time limit: 1.00 s
  • Memory limit: 512 MB
A company has $n$ employees, who form a tree hierarchy where each employee has a boss, except for the general director.

Your task is to process $q$ queries of the form: who is the lowest common boss of employees $a$ and $b$ in the hierarchy?

Input

The first input line has two integers $n$ and $q$: the number of employees and queries. The employees are numbered $1,2,\dots,n$, and employee $1$ is the general director.

The next line has $n-1$ integers $e_2,e_3,\dots,e_n$: for each employee $2,3,\dots,n$ their boss.

Finally, there are $q$ lines describing the queries. Each line has two integers $a$ and $b$: who is the lowest common boss of employees $a$ and $b$?

Output

Print the answer for each query.

Constraints
  • $1 \le n,q \le 2 \cdot 10^5$
  • $1 \le e_i \le i-1$
  • $1 \le a,b \le n$
Example

Input:
5 3
1 1 3 3
4 5
2 5
1 4


Output:
3
1
1