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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