Fence

M-O is a small robot that cleans the floor of the Axiom, a starliner built by the Buy n Large corporation. Tonight the great hall of the Axiom is a mess. Foreign microbes have spread over its floor, and wiping all of them is M-O's job.

The floor of the great hall is an infinite grid of $1 \times 1$ cells. The four edges and the two diagonals of every cell are drawn as white segments. Every microbe sits on a corner of a cell.

Cleaning takes a long time, so M-O first builds a fence around the microbes to stop them from spreading to other areas of the ship. M-O places the fence only on white segments, that is, on cell edges and cell diagonals. The fence is a single closed path, and every microbe must lie inside the fence or on the fence itself.

Given the positions of the microbes, find the perimeter of a shortest fence.

The input holds several test cases. Each test case starts with a line holding the number of microbes in the great hall, $n$ ($1 \le n \le 10000$). Each of the next $n$ lines holds two integers $x_i$ and $y_i$, the coordinates of the $i$-th microbe. The absolute value of every coordinate is at most $10^6$. The origin is an arbitrary cell corner. Two microbes are allowed to sit on the same corner.

The last line of the input holds a single $0$. That line is not a test case.

The perimeter of any fence laid on grid edges and grid diagonals can be written uniquely as $a + b\sqrt{2}$, where $a$ and $b$ are non-negative integers. For each test case, write the two integers $a$ and $b$ on one line, separated by a space, where $a + b\sqrt{2}$ is the perimeter of a shortest fence.

The fence is a closed path. If two parts of the fence overlap, the overlapping part counts twice.