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.

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).

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)

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

见题面 Input 部分

见题面 Output 部分