Problem Statement

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.

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 $(10^9+7)$.

Sample Input 1

3
2 3 1 4

Sample Output 1

121788

The following drawings show two valid cycles (but there are more!) with products $2 \\cdot 5 \\cdot 6 \\cdot 3 \\cdot 1 \\cdot 2 \\cdot 5 \\cdot 4 = 7200$ and $1 \\cdot 3 \\cdot 7 \\cdot 7 \\cdot 3 \\cdot 1 \\cdot 2 \\cdot 5 \\cdot 4 \\cdot 2 = 35280$. The third cycle is invalid because it doesn't have exactly two special edges.

Sample Input 2

2
1 2

Sample Output 2

36

The only simple cycle in the graph indeed has two special edges, and the product of vertices is $1 \\cdot 3 \\cdot 3 \\cdot 1 \\cdot 2 \\cdot 2 = 36$.

Sample Input 3

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

Sample Output 3

10199246