Hurray! They're integer roots.

Level pending

Let \(B\) be the set of all integers. Given \(2\) polynomials

\[p(x)=x^3+ax^2+bx+c\]

with \(a,b,c\) are elements of \(B\) and

\[q(x)=x^2-40x+2410\]

Assuming that \(p(x)=0\) has \(3\) distinct integer roots, \(p(2005)=-2005\) and \(p(q(x))=0\) doesn't have any real roots, what are the last \(3\) digits of the absolute value of the sum of all possible values of \(a\)?

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