Guarding the Farm

Farmer John's farm has many hills, and he wants to place one guard on top of each hill to keep his valuable milk-cows safe. Help him find how many guards he needs — that is, how many hilltops appear on the map.

The map is given as an integer matrix with $N$ rows and $M$ columns ($1 < N \le 700$, $1 < M \le 700$). Each entry is an altitude $H_{ij}$ with $0 \le H_{ij} \le 10000$.

A hilltop is a group of one or more adjacent cells that all share the same value, such that the group is bounded exclusively by the edge of the map or by cells of lower (smaller) altitude. Two cells are adjacent if the absolute difference of their row coordinates is at most $1$ and the absolute difference of their column coordinates is at most $1$ (that is, the eight orthogonal and diagonal neighbors).

• Line 1: Two space-separated integers $N$ and $M$.
• Lines $2 \ldots N+1$: Line $i+1$ describes row $i$ of the matrix as $M$ space-separated integers $H_{ij}$.

• A single integer: the number of hilltops.

In the sample input there are three hilltops: the cell of altitude $4$ in the top-left, one of the cells of altitude $2$ in the lower part, and the cell of altitude $1$ in the top-right corner.