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
#include <bits/stdc++.h>
#include <random>

using namespace std;

typedef unsigned long long ull;
typedef long long ll;
typedef double ld;
//#define ll __int128
//#define int ll
//#define char ll
//#define double ld
typedef vector<char> vc;
typedef vector<vc> vvc;
typedef vector<vvc> vvvc;
typedef pair<int, int> pii;
typedef pair<pii, pii> piii;
typedef pair<ll, ll> pll;
typedef vector<int> vi;
typedef vector<pii> vpi;
typedef vector<vi> vvi;
typedef vector<vvi> vvvi;
typedef vector<short> vs;
typedef vector<vs> vvs;
typedef vector<vvs> vvvs;
typedef vector<ll> vl;
typedef vector<vl> vvl;
typedef vector<vvl> vvvl;
typedef vector<ld> vld;
typedef vector<vld> vvld;
typedef vector<vvld> vvvld;
typedef vector<string> vst;
typedef vector<vst> vvst;
typedef pair<ld, ld> pld;

#define inmin(a, b) a = min(a, (b))
#define inmax(a, b) a = max(a, (b))
#define ALL(a) a.begin(),a.end()
#define RALL(a) a.rbegin(),a.rend()
#define sqr(x) ((x) * (x))
#define fori(i, n) for(int i = 0; i < int(n); ++i)
#define SZ(a) ((int)((a).size()))
#define triple(T) tuple<T, T, T>
#define quad(T) tuple<T, T, T, T>
#define watch(x) cerr << (#x) << " = " << (x) << endl;

#ifdef MAX_HOME
#define cerr cout
#else
#define cerr if (false) cerr
#endif

const double PI = 2 * acos(0.0);
#define rand shittttty_shit
mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());
mt19937_64 rng_64(chrono::steady_clock::now().time_since_epoch().count());

const string DIGITS = "0123456789";
const string ALPH = "abcdefghijklmnopqrstuvwxyz";


template<class T0, class T1>
inline ostream &operator<<(ostream &out, pair<T0, T1> &a) {
    return out << "{" << a.first << ", " << a.second << "}";
}

template<class T0, class T1>
inline istream &operator>>(istream &in, pair<T0, T1> &a) {
    return in >> a.first >> a.second;
}

template<class T0, class T1, class T2>
inline ostream &operator<<(ostream &out, tuple<T0, T1, T2> &a) {
    return out << "{" << get<0>(a) << ", " << get<1>(a) << ", " << get<2>(a) << "}";
}

template<class T0, class T1, class T2, class T3>
inline ostream &operator<<(ostream &out, tuple<T0, T1, T2, T3> &a) {
    return out << "{" << get<0>(a) << ", " << get<1>(a) << ", " << get<2>(a) << ", " << get<3>(a) << "}";
}

template<class T>
inline ostream &operator<<(ostream &out, vector<T> &a) {
    out << "[";
    fori (i, a.size())out << a[i] << vector<string>{", ", "]  "}[i + 1 == a.size()];
    return out;
}


void smain();


signed main() {
    ios::sync_with_stdio(0);
    cin.tie(0);
    cout.tie(0);
#ifdef MAX_HOME
    freopen("input.txt", "r", stdin);
    clock_t start = clock();
#endif
#ifdef HK_CUP
    freopen("finput.txt", "r", stdin);
    freopen("output.txt", "w", stdout);
#endif

    cout << setprecision(12) << fixed;
    smain();
#ifdef MAX_HOME
#ifndef HK_CUP
    cout << "\n\nTOTAL EXECUTION TIME: " << float(clock() - start) / CLOCKS_PER_SEC << endl;
#endif
#endif
    return 0;
}

const int M = 1e9 + 7;

int pw(int a, int n = M - 2) {
    int ret = 1;
    while (n) {
        if (n & 1)
            ret = (ll) ret * a % M;
        a = (ll) a * a % M;
        n >>= 1;
    }
    return ret;
}

int h, l;
vi p;

pii merge_pairs(const pii &a, const pii &b) {
    return {(a.first + b.first) % M, (a.second + b.second) % M};
}

int ans = 0;

void insum(int &a, int b) {
    a = (a + b) % M;
}

vector<pair<int, pii>> my_merge(const vector<pair<int, pii>> &a, const vector<pair<int, pii>> &b, int v) {
    vector<pair<int, pii>> ret;
    int ai = 0, bi = 0;
    while (ai < a.size() && bi < b.size()) {
        if (a[ai].first == b[bi].first) {
            int addition = ((ll) a[ai].second.first * b[bi].second.second + (ll) a[ai].second.second * b[bi].second.first) % M;
            addition = (ll) addition * v % M * a[ai].first % M;
            insum(ans, addition);
            ret.push_back({a[ai].first, merge_pairs(a[ai].second, b[bi].second)});
            ai++;
            bi++;
        } else if (a[ai].first < b[bi].first) {
            ret.push_back(a[ai]);
            ai++;
        } else {
            ret.push_back(b[bi]);
            bi++;
        }
    }
    while (ai < a.size()) {
        ret.push_back(a[ai++]);
    }
    while (bi < b.size()) {
        ret.push_back(b[bi++]);
    }
    return ret;
}


vector<pair<int, pii>> dfs(int v) {
    if (v >= l) {
        /* leaf */;
        vector<pair<int, pii>> ret;
        int u = p[v - l];
        int x = v;
        while (u != 1) {
            x = (ll) x * u % M;
            pii val = {0, 0};
            if (u & 1)
                val.second = x;
            else
                val.first = x;
            u >>= 1;
            ret.push_back({u, val});
        }
        reverse(ALL(ret));
        return ret;
    }
    vector<pair<int, pii>> f = dfs(v << 1);
    vector<pair<int, pii>> s = dfs(v << 1 | 1);
    vector<pair<int, pii>> ret = my_merge(f, s, v);
    for (auto &x : ret) {
        x.second.first = (ll) x.second.first * v % M;
        x.second.second = (ll) x.second.second * v % M;
    }
    return ret;
}

void smain() {
    cin >> h;
    l = 1 << (h - 1);
    p.resize(l);
    fori (i, l) {
        cin >> p[i];
        p[i]--;
        p[i] += l;
    }

    dfs(1);
    cout << ans;
}