Problem Statement

Takahashi is in charge of the management system for a company consisting of N employees. The employees are numbered 1 through N.

The company has a clear hierarchy. Employee 1 is the president, and each employee i (2 \leq i \leq N) other than employee 1 has exactly one direct superior P_i. This hierarchy forms a rooted tree with employee 1 as the root.

The distance \mathrm{dist}(u, v) between employee u and employee v is defined as the number of edges on the unique simple path between u and v on the rooted tree. In particular, \mathrm{dist}(u, u) = 0.

The depth \mathrm{depth}(v) of employee v is defined as the distance from the root, \mathrm{dist}(1, v).

Employee v is said to be an ancestor of employee u if v can be reached by starting from u and repeatedly moving to the direct superior 0 or more times toward the root (employee 1). By this definition, employee u themself (0 steps) and employee 1 (tracing all the way to the root) are always ancestors of u.

The lowest common ancestor (LCA) \mathrm{LCA}(S) of a non-empty set of employees S is defined as the employee with the maximum depth (farthest from the root) among all common ancestors of the employees in S. By the properties of rooted trees, such an employee is uniquely determined. If the set contains only one employee, \mathrm{LCA}(\{u\}) = u.

Takahashi needs to process Q "meeting summons queries".

In the j-th query (1 \leq j \leq Q), K_j distinct employees V_{j,1}, V_{j,2}, \ldots, V_{j,K_j} are summoned to a meeting. The facilitator of the meeting is the LCA of the summoned K_j employees. Note that the facilitator may or may not be among the summoned members.

For each query, find Value 1 and Value 2 described below.

Value 1: Find the sum of distances from the facilitator r = \mathrm{LCA}(\{V_{j,1}, V_{j,2}, \ldots, V_{j,K_j}\}) to each of the K_j summoned employees. That is, compute

\sum_{i=1}^{K_j} \mathrm{dist}(r, V_{j,i})

(When K_j = 1, r = V_{j,1}, so Value 1 is 0.)

Value 2: Consider having exactly 1 of the K_j summoned employees be absent and holding the meeting with the remaining K_j - 1 employees. Let r' be the LCA of the remaining K_j - 1 employees as the new facilitator, and compute the sum of distances from r' to each of the remaining K_j - 1 employees. Find the minimum possible sum of distances over all choices of the absent employee (K_j ways).

Letting S_j = \{V_{j,1}, V_{j,2}, \ldots, V_{j,K_j}\}, the value to find is

\min_{1 \leq t \leq K_j} \sum_{\substack{1 \leq i \leq K_j \\ i \neq t}} \mathrm{dist}\!\left(\mathrm{LCA}\!\left(S_j \setminus \{V_{j,t}\}\right),\, V_{j,i}\right)

Note that the facilitator r' of the remaining K_j - 1 employees may or may not be included in the remaining members.

However, if K_j = 1, leaving 1 person out results in an empty set and the LCA cannot be defined, so output -1 as Value 2. When K_j \geq 2, no matter who is absent, the remaining set is non-empty, so the LCA is always uniquely determined.

Constraints

• 1 \leq N \leq 2 \times 10^5
• 1 \leq Q \leq 2 \times 10^5
• 1 \leq P_i \leq i - 1 (2 \leq i \leq N)
• 1 \leq K_j \leq N (1 \leq j \leq Q)
• 1 \leq V_{j,k} \leq N (1 \leq k \leq K_j)
• V_{j,1}, V_{j,2}, \ldots, V_{j,K_j} are all distinct in each query
• \displaystyle \sum_{j=1}^{Q} K_j \leq 2 \times 10^5
• All input values are integers.

Sample Input 1

7 4
1 1 2 2 3 3
3 4 5 2
2 4 6
1 7
4 4 5 6 7

Sample Output 1

2 1
4 0
0 -1
8 6

Sample Input 2

5 5
1 2 3 4
1 3
2 2 5
3 1 4 5
1 1
4 2 3 4 5

Sample Output 2

0 -1
3 0
7 1
0 -1
6 3

Sample Input 3

15 8
1 1 2 2 3 3 4 4 5 6 6 7 10 10
5 8 9 14 15 10
4 11 12 13 7
6 4 5 6 7 14 15
3 2 8 9
7 1 3 6 11 12 13 15
2 14 15
1 5
8 8 9 10 11 12 13 14 15

Sample Output 3

12 9
7 5
16 12
4 2
16 12
2 0
0 -1
26 22

Sample Input 4

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

Sample Output 4

34 30
19 16
9 5
9 6
12 6
26 22
2 0
0 -1
21 18
28 24
5 2
6 4

Sample Input 5

1 3

1 1
1 1
1 1

Sample Output 5

0 -1
0 -1
0 -1