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 $(10^9+7)$.