Discussion Club 2

We visit the Bajtocki Discussion Club again. The club has $2^n$ members, and each member has stated their views on $n$ fundamental questions. The exact wording of the questions does not matter. Each question has one of two possible answers (for example, coffee or tea?). A person's views can be encoded as a string of bits, which read as a binary number gives an integer from 0 to $2^n-1$. No two club members have the same views.

Today the club meets, and $m$ members attend. Each has already taken a seat at a round table. The members want to split into two teams to prepare for the main topic in advance. To keep things orderly, each team must consist of members sitting in consecutive seats around the table. For a productive discussion, every team must cover all views: for each fundamental question and each possible answer, if some member of the first team holds that view, then some member of the second team must also hold it.

Write a program that counts the number of ways to split the members into two teams.

The first line contains two integers $n$ and $m$ ($n \ge 2$, $m \ge 3$), the number of fundamental questions and the number of members at the meeting. The second line contains $m$ pairwise distinct integers from 0 to $2^n-1$, describing the views of the members in order around the table.

Print one integer, the number of ways to split the members into two teams that satisfy the conditions.

$n \le 30$

$m \le \min(2^n, 2{,}000{,}000)$