Bro , do you even Euler?

Number Theory Level 4

\[ \begin{eqnarray} a^3-b^3-c^3&=&3abc \\ a^2 &=& 2(b+c) \\ \end{eqnarray} \]

Find the triplets \((a,b,c) \in \mathbb {Z^{+}}\) that satisfy the system of equations above.

Details and assumptions:

  • Input the sum of all possible numbers in the triplets as an answer. As an explicit example, if \((1,2,3)\) is the only triplet, input 6 but if \((2,3,4)\) is also an answer input 15.

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