CF1761C.Set Construction

You are given a binary matrix $b$ (all elements of the matrix are $0$ or $1$ ) of $n$ rows and $n$ columns.

You need to construct a $n$ sets $A_1, A_2, \ldots, A_n$ , for which the following conditions are satisfied:

• Each set is nonempty and consists of distinct integers between $1$ and $n$ inclusive.
• All sets are distinct.
• For all pairs $(i,j)$ satisfying $1\leq i, j\leq n$ , $b_{i,j}=1$ if and only if $A_i\subsetneq A_j$ . In other words, $b_{i, j}$ is $1$ if $A_i$ is a proper subset of $A_j$ and $0$ otherwise.

Set $X$ is a proper subset of set $Y$ , if $X$ is a nonempty subset of $Y$ , and $X \neq Y$ .

It's guaranteed that for all test cases in this problem, such $n$ sets exist. Note that it doesn't mean that such $n$ sets exist for all possible inputs.

If there are multiple solutions, you can output any of them.

Each test contains multiple test cases. The first line contains a single integer $t$ ( $1\le t\le 1000$ ) — the number of test cases. The description of test cases follows.

The first line contains a single integer $n$ ( $1\le n\le 100$ ).

The following $n$ lines contain a binary matrix $b$ , the $j$ -th character of $i$ -th line denotes $b_{i,j}$ .

It is guaranteed that the sum of $n$ over all test cases does not exceed $1000$ .

It's guaranteed that for all test cases in this problem, such $n$ sets exist.

For each test case, output $n$ lines.

For the $i$ -th line, first output $s_i$ $(1 \le s_i \le n)$ — the size of the set $A_i$ . Then, output $s_i$ distinct integers from $1$ to $n$ — the elements of the set $A_i$ .

If there are multiple solutions, you can output any of them.

It's guaranteed that for all test cases in this problem, such $n$ sets exist.

In the first test case, we have $A_1 = \{1, 2, 3\}, A_2 = \{1, 3\}, A_3 = \{2, 4\}, A_4 = \{1, 2, 3, 4\}$ . Sets $A_1, A_2, A_3$ are proper subsets of $A_4$ , and also set $A_2$ is a proper subset of $A_1$ . No other set is a proper subset of any other set.

In the second test case, we have $A_1 = \{1\}, A_2 = \{1, 2\}, A_3 = \{1, 2, 3\}$ . $A_1$ is a proper subset of $A_2$ and $A_3$ , and $A_2$ is a proper subset of $A_3$ .