2014-philic Cubic

Algebra Level 5

Let \[P(x)=x^3-(2014+2\sqrt{2014})x+4028\]

If the smallest root of \(P(x)\) is \(r_1\), and the other two roots are \(r_2\) and \(r_3\), then \(r_2\times r_3\) can be expressed as \[a+\sqrt{b+c\sqrt{d}}\] for integers \(a, b, c, d\) with \(d\) square-free. Find \[a+b+c+d\pmod{10000}\]

Note: if you are going to use Wolfram Alpha, don't even bother answering the problem (cheater!)

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