Twin Binary Trees

Inspired by the tv series Stranger Things, bear Limak is going for a walk between two mirror worlds.

There are two perfect binary trees of height H, each with the standard numeration of vertices from 1 to 2^H-1. The root is 1 and the children of x are 2 \cdot x and 2 \cdot x + 1.

Let L denote the number of leaves in a single tree, L = 2^{H-1}.

You are given a permutation P_1, P_2, \ldots, P_L of numbers 1 through L. It describes L special edges that connect leaves of the two trees. There is a special edge between vertex L+i-1 in the first tree and vertex L+P_i-1 in the second tree.

graph for sample1

drawing for the first sample test, permutation P = (2, 3, 1, 4), special edges in green

Let's define product of a cycle as the product of numbers in its vertices. Compute the sum of products of all simple cycles that have exactly two special edges, modulo (10^9+7).

A simple cycle is a cycle of length at least 3, without repeated vertices or edges.

Constraints

• 2 \leq H \leq 18
• 1 \leq P_i \leq L where L = 2^{H-1}
• P_i \neq P_j (so this is a permutation)

Input

Input is given from Standard Input in the following format (where L = 2^{H-1}).

H P_1 P_2 \cdots P_L

Output

Compute the sum of products of simple cycles that have exactly two special edges. Print the answer modulo (10^9+7).

Examples

Input

3 2 3 1 4

Output

121788

Input

2 1 2

Output

36

Input

5 6 14 15 7 12 16 5 4 11 9 3 10 8 2 13 1

Output

10199246

inputFormat

Input

Input is given from Standard Input in the following format (where L = 2^{H-1}).

H P_1 P_2 \cdots P_L

outputFormat

Output

Compute the sum of products of simple cycles that have exactly two special edges. Print the answer modulo (10^9+7).

Examples

Input

3 2 3 1 4

Output

121788

Input

2 1 2

Output

36

Input

5 6 14 15 7 12 16 5 4 11 9 3 10 8 2 13 1

Output

10199246

样例

2
1 2

36