# Dogfight

Two players play a game on the cartesian plane which occurs in 2 phases. In the first phase, a number $$n$$ is selected from $$\{2,3,4,\ldots, 999\}.$$ The first player then places a point at a location $$(x,y)$$ in the plane satisfying $$-n \leq x \leq n, -n \leq y \leq n$$ and $$x,y$$ integers. The players alternate placing points until a total of $$n$$ points have been placed.

In the second phase, the first player picks two distinct points in the plane that are not joined and joins them by a curve that does not intersect itself, any of the other $$n-2$$ points, or any of the already drawn curves. The players alternate turns, drawing curves in this manner. The first player who is unable to draw a curve loses. For the 998 starting values of $$n,$$ determine how many of these the first player has a winning strategy for.

Details and assumptions

The players get to choose where they want to place their points.

The curve is a continuous path which does not include the endpoints. Hence, 2 curves may seem to intersect at one of the $$n$$ points.

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