# Sinful sum

Algebra Level 5

Let $$N$$ be the number of ordered quadruples $$(a,b,c,d)$$ of integers, each of which are from $$-10$$ to $$10$$ (inclusive), such that $\begin{cases} a+b+c+d & =0\\ \sin a + \sin b+ \sin c +\sin d &=0. \end{cases}$ What are the last three digits of $$N?$$

Details and assumptions

The calculations are done in radians, not degrees.

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