Enclosing the Grid Points

Peter and Bob play a game on a sheet of graph paper. Peter marks a few grid nodes with points, and Bob draws a polygon around them. Every marked node has to lie strictly inside the polygon, never on its border. Every side of the polygon lies along a side or a diagonal of a grid cell, and the perimeter is as short as possible. Compute that perimeter.

The first line contains the number of points Peter marked, $N$ ($1 \le N \le 100\,000$). Each of the next $N$ lines contains two integers $x_i$ and $y_i$, the coordinates of one point. Each coordinate has absolute value at most $10^6$. Several points can share the same coordinates.

Print the perimeter of the polygon on one line, rounded to exactly six digits after the decimal point.