AMC 12 2014a Problem 25

The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coordinates is it true that $|4x+3y|\leq 1000$?

$\textbf{(A) }38 \textbf{(B) }40 \textbf{(C) }42 \textbf{(D) }44 \textbf{(E) }46$