Copy of a Copy of a Copy
Contest: Codeforces Round 839 (Div. 3)
Difficulty rating: 2000
Tags: constructive algorithms, dfs and similar, graphs, implementation, sortings

It all started with a black-and-white picture, that can be represented as an $$$n \times m$$$ matrix such that all its elements are either $$$0$$$ or $$$1$$$. The rows are numbered from $$$1$$$ to $$$n$$$, the columns are numbered from $$$1$$$ to $$$m$$$.

Several operations were performed on the picture (possibly, zero), each of one of the two kinds:

- choose a cell such that it's not on the border (neither row $$$1$$$ or $$$n$$$, nor column $$$1$$$ or $$$m$$$) and it's surrounded by four cells of the opposite color (four zeros if it's a one and vice versa) and paint it the opposite color itself;
- make a copy of the current picture.

Note that the order of operations could be arbitrary, they were not necessarily alternating.

You are presented with the outcome: all $$$k$$$ copies that were made. Additionally, you are given the initial picture. However, all $$$k+1$$$ pictures are shuffled.

Restore the sequence of the operations. If there are multiple answers, print any of them. The tests are constructed from the real sequence of operations, i. e. at least one answer always exists.

Input format:
The first line contains three integers $$$n, m$$$ and $$$k$$$ ($$$3 \le n, m \le 30$$$; $$$0 \le k \le 100$$$) — the number of rows and columns of the pictures and the number of copies made, respectively.

Then $$$k+1$$$ pictures follow — $$$k$$$ copies and the initial picture. Their order is arbitrary.

Each picture consists of $$$n$$$ lines, each consisting of $$$m$$$ characters, each character is either $$$0$$$ or $$$1$$$. There is an empty line before each picture.

Output format:
In the first line, print a single integer — the index of the initial picture. The pictures are numbered from $$$1$$$ to $$$k+1$$$ in the order they appear in the input.

In the second line, print a single integer $$$q$$$ — the number of operations.

Each of the next $$$q$$$ lines should contain an operation. The operations should be listed in order they were applied. Each operation is one of two types:

- $$$1$$$ $$$x$$$ $$$y$$$ — recolor a cell $$$(x, y)$$$ (the $$$y$$$-th cell in the $$$x$$$-th row, it should not be on the border and it should be surrounded by four cells of opposite color to itself);
- $$$2$$$ $$$i$$$ — make a copy of the current picture and assign it index $$$i$$$ (picture with index the $$$i$$$ should be equal to the current picture).

Each index from $$$1$$$ to $$$k+1$$$ should appear in the output exactly once — one of them is the index of the initial picture, the remaining $$$k$$$ are arguments of the operations of the second kind.

If there are multiple answers, print any of them. The tests are constructed from the real sequence of operations, i. e. at least one answer always exists.

Constraints:
Time limit: 2.0 s
Memory limit: 256.0 MB

Example input:
3 3 1

010
111
010

010
101
010

Example output:
2
2
1 2 2
2 1

Example input:
4 5 3

00000
01000
11100
01000

00000
01000
10100
01000

00000
01010
10100
01000

00000
01000
10100
01000

Example output:
3
5
1 2 4
2 2
2 4
1 3 2
2 1

Editorial:
Notice the following: once you apply the recolor operation to some cell, you can never recolor it again. That happens because you can't recolor its neighbors too as each of them has at least one neighbor of the same color — this cell itself.
In particular, that implies that applying a recolor operation always decreases the possible number of operations that can be made currently. It doesn't always decrease them by $$$1$$$: from $$$1$$$ to $$$5$$$ operations can become unavailable, but it always decreases.
That gives us an order of copies. Just sort them in the decreasing order of the number of recolor operations that can be made currently. If the numbers are the same, the copies must be equal, so their order doesn't matter.
The only thing remains is to apply the operations. Turns out, their order doesn't matter at all. Consider all different cells for a pair of adjacent pictures. It's never possible that there are two different cells that are adjacent to each other. Thus, no operation can interfere with another one. Just print all positions of different cells in any order you want and make a copy.
Overall complexity: $$$O(nmk + k \log k)$$$.