Closed room mystery

Task

Given a number $N$ and a matrix $F$ of dimensions $N \times N$, we define a set as closed if it is non-empty and for any two elements $x$ and $y$ in the set (not necessarily distinct!), $F_{x,y}$ is also in the set. The task is to display all subsets of the set $\{1, 2, \ldots, N\}$ that are closed and do not strictly contain any other closed set.

Input data

The first line contains the number $N$, followed by the matrix $F$ on the subsequent lines.

Output data

$X$, the number of such sets, followed by each set on a new line, sorted in ascending order by the minimum element, and in case of a tie, by size. Additionally, the individual sets should also be sorted in increasing order.

Constraints and clarifications

• $1 \leq N \leq 1500$
• The values in matrix $F$ are natural numbers between $1$ and $N$
• A set $A$ strictly contains another set $B$ if every element of $B$ belongs to $A$ and $A \neq B$

Subtasks

• For $30$ points, $N \leq 500$
• For the remaining $70$ points, there are no additional constraints.
• It can be proven that the required sets can be uniquely sorted according to the given criteria.
• It can be proven that, under the given conditions, the sum of the sizes of the required sets does not exceed $10^6$

Example 1

stdin

5
5 4 1 5 4
5 1 2 4 4
5 3 3 3 2
4 5 5 2 5
2 5 4 4 2

stdout

2
1 2 4 5
3

Explanation

The sets $\{1, 2, 4, 5\}$ and $\{3\}$ do not strictly contain any other closed set. Note that we have not displayed the set $\{1, 2, 3, 4, 5\}$ although it is closed because it contains the closed set $\{3\}$.