Farmer John wishes to build a corral for his cows. Being finicky beasts, they demand that the corral be square and that the corral contain at least C (1 <= C <= 500) clover fields for afternoon treats. The corral's edges must be parallel to the X,Y axes. FJ's land contains a total of N (C <= N <= 500) clover fields, each a block of size 1 x 1 and located at with its lower left corner at integer X and Y coordinates each in the range 1..10,000. Sometimes more than one clover field grows at the same location; such a field would have its location appear twice (or more) in the input. A corral surrounds a clover field if the field is entirely located inside the corral's borders. Help FJ by telling him the side length of the smallest square containing C clover fields.


Line 1: Two space-separated integers: C and N Lines 2..N+1: Each line contains two space-separated integers that are the X,Y coordinates of a clover field. The input is terminated with N = K = 0.


Line 1: A single line with a single integer that is length of one edge of the minimum size square that contains at least C clover fields.

Sample Input

3 4 1 2 2 1 4 1 5 2

Sample Output



Explanation of the sample: |* * | * * +------ Below is one 4x4 solution (C's show most of the corral's area); many others exist. |CCCC |CCCC |*CCC* |C*C* +------


USACO 2006 January Gold