An Interactive Problem

The jury is in trouble. NWERC is tomorrow, and the problem set still needs a good interactive problem. All $n^2$ jury members have an idea, and after hours of arguing they cannot decide which idea is the nicest. They decide to draw lots.

Each jury member draws a lot from a collection of unique numbers. The lots lie face-down on a table, arranged in the shape of a square. You are the independent lot-drawer. You cannot see the table, so you pick lots by shouting coordinates. The jury is in a hurry, so you must find the lot with the highest number in at most $n^2 + 100$ commands. You do not know the size of the square in advance, but you know $1 \le n \le 100$.