Let $m$ be the number of integers $n$, $1 \le n \le 2005$, such that the polynomial $x^{2n} + 1 + (x + 1)^{2n}$ is divisible by $x^2 + x + 1$. find the remainder when $m$ is divided by 1000.

If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. which set of triangles shows the similar triangles created by the altitude in the diagram?