Recover Polygon (hard)

Description:

Input:

The input contains multiple test cases.

The first line of each test case contains two space-separated integers N and M, where N is the size of the lattice grid (5 ≤ N ≤ 100000) and M is the number of lattice points for which the Zombie Contamination level is 1, 2, or 3 (8 ≤ M ≤ 200000).

The second line of each test case contains M pairs of integers x₁, y₁, ..., x_M, y_M – coordinates of the cells with Zombie Contamination level not equal to 0 nor 4. It is guaranteed that 1 ≤ x_i, y_i ≤ N. All pairs x_i, y_i are different.

Cells are enumerated based on the coordinates of their upper right corner. This means that the bottommost leftmost cell that touches the origin has coordinates (1, 1), and the uppermost leftmost cell is identified as (1, N).

The last line of the file contains two zeroes. This line should not be treated as a test case. The sum of the M values for all tests in one file will not exceed 200000.

Output:

For each test case, the following output is expected:

The first line of the output should contain one integer V, the number of vertices of the polygon that is the secret lair. The next V lines each should contain two integers, denoting the vertices of the polygon in the clockwise order, starting from the lexicographically smallest vertex.

Sample Input:

8 19
2 3 2 4 2 5 3 3 3 5 4 3 4 5 4 6 5 2 5 3 5 6 6 2 6 3 6 4 6 5 6 6 6 7 7 6 7 7
5 8
2 2 2 3 2 4 3 2 3 4 4 2 4 3 4 4
0 0

Sample Output:

4
2 3
2 4
6 6
5 2
4
2 2
2 3
3 3
3 2

Note:

It is guaranteed that the solution always exists and is unique. It is guaranteed that in the correct solution the coordinates of the polygon vertices are between 1 and N - 1. A vertex (x₁, y₁) is lexicographically smaller than vertex (x₂, y₂) if x₁ < x₂ or .