Recently, Angie's math teacher began teaching the class about trees (the graph theory kind, of course). As homework, he gave everyone a rooted tree of ~N~ nodes (the root node is ~1~) and asked them to do ~Q~ LCA queries on the tree (each query is in the form x y, and its answer is the LCA of nodes ~x~ and ~y~ in the tree). Since he didn't have the funds to make proper test data for his trees, he generated them randomly with the following algorithm: for every node ~i~ ~(2 \le i \le N)~, he picked a uniform random integer in the range ~[1, i)~ to be its parent node.

Angie is feeling very demotivated and doesn't want to do her work. Can you solve the queries for her?

Constraints

~2 \le x, y \le N \le 6 \times 10^6~

~1 \le Q \le 10^6~

~1 \le p_i < i~ for all ~2 \le i \le N~

Input Specification

The first line contains the integers ~N~ and ~Q~.

The second line contains ~N-1~ integers ~p_2, p_3, \dots, p_N~, the parents of nodes ~2, 3, \dots, N~ respectively. Note that node ~1~ does not have a parent as it's the root node of the tree.

The next ~Q~ lines each contain a query of the form x y.

Output Specification

For each query, output its answer on a new line.

Sample Input

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

Sample Output

1
1
4
1
2

Explanation

The tree from the sample input: