Two Doors

Dodo set out to find Ea, who was swept away by a storm across space and time. Dodo knows only that Ea is between $-n$ years and $n$ years in the future from now. While searching for a time machine, Dodo learned about a strange door from the evil Black Wizard.

The door has a front side and a back side. The front side is always 1 year ahead of the back side. In other words, going through the door from front to back takes you 1 year into the past, and going through from back to front takes you 1 year into the future.

The devilish Black Wizard would not simply give Dodo the door. Instead, he offered a passage made of $n$ doors joined together. After long study, Dodo installed $m$ switches, each of which turns a specific combination of doors 180 degrees. Now let us use these switches to rescue Ea!

The first line contains $n$ ($1 \le n \le 100$) and $m$ ($1 \le m \le 20$).

The second line contains a string of length $n$ made only of 0 and 1. The $j$-th character gives the initial state of the $j$-th door. 0 means the front side faces Dodo, and 1 means the back side faces Dodo.

From the third line to line $m+2$, there are $m$ strings of length $n$ made only of 0 and 1. The $j$-th character of the $(i+2)$-th line tells whether switch $i$ turns door $j$ 180 degrees. 0 means it does not turn the door, and 1 means it does.

Print $(2n+1)$ lines in total. For every $-n \le i \le n$, print the answer for trying to go $i$ years into the future on line $(i+n+1)$.

On each line, if a solution exists, print the number $k$ ($0 \le k \le m$) of switches to press, followed by the switch numbers separated by spaces. If no solution exists, print -1.

If there are several combinations of switches to press, print any one of them.

In the figure, red means the front side of a door faces you and blue means the back side faces you. The upper passage in the figure is the initial state. Passing through this passage takes you $-3$ years into the future, that is, 3 years into the past. Pressing switch 2 changes the passage to the lower one. Passing through that passage takes you 1 year into the future.