A11075 | Random Elections

The presidential election is coming in Bearland next year! Everybody is so excited about this!

So far, there are three candidates, Alice, Bob, and Charlie.

There are $n$ citizens in Bearland. The election result will determine the life of all citizens of Bearland for many years. Because of this great responsibility, each of $n$ citizens will choose one of six orders of preference between Alice, Bob and Charlie uniformly at random, independently from other voters.

The government of Bearland has devised a function to help determine the outcome of the election given the voters preferences. More specifically, the function is  (takes $n$ boolean numbers and returns a boolean number). The function also obeys the following property: $f(1-x_{1},1-x_{2},...,1-x_{n})=1-f(x_{1},x_{2},...,x_{n})$ .

Three rounds will be run between each pair of candidates: Alice and Bob, Bob and Charlie, Charlie and Alice. In each round, $x_{i}$ will be equal to $1$ , if $i$ -th citizen prefers the first candidate to second in this round, and $0$ otherwise. After this, $y=f(x_{1},x_{2},...,x_{n})$ will be calculated. If $y=1$ , the first candidate will be declared as winner in this round. If $y=0$ , the second will be the winner, respectively.

Define the probability that there is a candidate who won two rounds as $p$ . $p·6^{n}$ is always an integer. Print the value of this integer modulo $10^{9}+7=1\ 000\ 000\ 007$ .

The first line contains one integer $n$ ( $1<=n<=20$ ).

The next line contains a string of length $2^{n}$ of zeros and ones, representing function $f$ . Let $b_{k}(x)$ the $k$ -th bit in binary representation of $x$ , $i$ -th (0-based) digit of this string shows the return value of $f(b_{1}(i),b_{2}(i),...,b_{n}(i))$ .

It is guaranteed that $f(1-x_{1},1-x_{2},...,1-x_{n})=1-f(x_{1},x_{2},...,x_{n})$ for any values of $x_{1},x_{2},ldots,x_{n}$ .

Output one integer — answer to the problem.