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Geometry Level 5

\[\large x^{3}\,+\,\left(\sqrt[b]{\frac{b \sqrt[b]{d} - c}{a}}\right) x^{2}\,+\, \left(\frac{b-\sqrt[b]{d}}{a^{\frac{a}{b}} \sqrt[b]{c-b \sqrt[b]{d}} }\right) x\,-\,\frac{1}{a}\] Let \(a,b,c\) and \(d\) be constant prime numbers such that the equation above has roots:

\[ \sqrt[3]{\sin\left(\dfrac{57\pi}{266}\right)}, \quad \sqrt[3]{\sin\left( -\dfrac{23\pi}{322}\right)}, \quad \sqrt[3]{\sin\left(-\dfrac{215\pi}{602}\right)} . \]

Find \(a+b+c+d\).

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