Forest Fire

A forest fire has broken out in the integer forest! The integer forest is an infinitely large two-dimensional plane with a tree at every point with integer coordinates. Right now $N$ of these trees are burning. Every minute, fire spreads from each burning tree to its four neighbors: the trees directly north, east, west, and south. To stop the fire, the fire brigade has cut down $M$ trees. A cut-down tree cannot catch fire, so these points act as a wall. You want to know how much damage the fire will cause. In $T$ minutes, a heavy rainstorm will put out the whole fire. So you want to know how many trees are burning after $T$ minutes.

The first line contains three integers: the number of burning trees $N$ ($1 \leq N \leq 100$), the number of cut-down trees $M$ ($0 \leq M \leq 100$), and the number of minutes $T$ until the heavy rain ($0 \leq T \leq 10^9$). The next $N$ lines each contain two integers $x_i, y_i$ ($0 \leq x_i, y_i < 10^5$), the coordinates of a burning tree. The next $M$ lines each contain two integers $x_i, y_i$ ($0 \leq x_i, y_i < 10^5$), the coordinates of a cut-down tree. No two trees, whether burning or cut down, share the same coordinates.

Print one integer, the number of trees that are burning after $T$ minutes.

In the first example, the fire spreads to $(0,2)$, $(1,3)$, and $(2,2)$. That makes 4 burning trees in total. In the second example, the only burning tree is surrounded by cut-down trees. So only one tree burns, even though $T$ is very large. In the third example, the fire does not spread because $T = 0$, so the answer is 4.