Load Balancing (Silver)

Farmer John's $n$ cows are standing at distinct locations (x_1, y_1), \ldots, (x_n, y_n) on his two-dimensional farm. FJ wants to partition his field by building a long (effectively infinite) north-south fence along the line x = a, where $a$ is an even integer, ensuring that the fence does not pass through the location of any cow. He also wants to build a long (effectively infinite) east-west fence along the line y = b, where $b$ is an even integer. These two fences intersect at the point $(a,b)$ and together divide the field into four regions.

FJ wants to choose $a$ and $b$ so that the cows in the four resulting regions are as evenly distributed as possible, with no region containing too many cows. Let $m$ be the maximum number of cows in any one of the four regions. FJ wants to make $m$ as small as possible. Please help him determine the minimum possible value of $m$.