A Tricky Diophantine Equation

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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