A15654 | Playoff

$2^n$ teams participate in a playoff tournament. The tournament consists of $2^n - 1$ games. They are held as follows: in the first phase of the tournament, the teams are split into pairs: team $1$ plays against team $2$ , team $3$ plays against team $4$ , and so on (so, $2^{n-1}$ games are played in that phase). When a team loses a game, it is eliminated, and each game results in elimination of one team (there are no ties). After that, only $2^{n-1}$ teams remain. If only one team remains, it is declared the champion; otherwise, the second phase begins, where $2^{n-2}$ games are played: in the first one of them, the winner of the game " $1$ vs $2$ " plays against the winner of the game " $3$ vs $4$ ", then the winner of the game " $5$ vs $6$ " plays against the winner of the game " $7$ vs $8$ ", and so on. This process repeats until only one team remains.

The skill level of the $i$ -th team is $p_i$ , where $p$ is a permutation of integers $1$ , $2$ , ..., $2^n$ (a permutation is an array where each element from $1$ to $2^n$ occurs exactly once).

You are given a string $s$ which consists of $n$ characters. These characters denote the results of games in each phase of the tournament as follows:

- if $s_i$ is equal to 0, then during the $i$ -th phase (the phase with $2^{n-i}$ games), in each match, the team with the lower skill level wins;
- if $s_i$ is equal to 1, then during the $i$ -th phase (the phase with $2^{n-i}$ games), in each match, the team with the higher skill level wins.

Let's say that an integer $x$ is winning if it is possible to find a permutation $p$ such that the team with skill $x$ wins the tournament. Find all winning integers.

The first line contains one integer $n$ ( $1 \le n \le 18$ ).

The second line contains the string $s$ of length $n$ consisting of the characters 0 and/or 1.

Print all the winning integers $x$ in ascending order.