Score : $1000$ points

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 $(10^9+7)$.

3
2 3 1 4

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.

2
1 2

36

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

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

10199246