Define a *growing spiral* in the plane as a sequence of points with integer coordinates $P_0=(0,0), P_1, \dots, P_n$ such that $n \ge 2$ and:

- The directed line segments $P_0P_1, P_1P_2, \dots, P_{n-1}P_n$ follow successive coordinate directions: east (for $P_0P_1$), north, west, south, east, etc.
- The lengths of these line segments are positive and strictly increasing.

**Question:**

How many of the points $(x,y)$ with integer coordinates $0 \le x \le 2011, 0 \le y \le 2011$ *cannot* be the last point, $P_n$, of any growing spiral?

Let's think step by step and output the final answer within \boxed{}.
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