A Tricky Diophantine Equation

Number Theory Level 5

Find the last three digits of the sum of all positive integers \(n \leq 2014\) for which there exist integers \(a, b, c, d, e\) (not necessarily distinct) such that \[a^2+b^2+c^2+d^2+e^2= \left( 3^n + 1 \right) ^2 \left( 2 \cdot 3^{2n} + 5 \right) .\]

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