Bro , do you even Euler?

Number Theory Level 4

$\begin{aligned} a^3-b^3-c^3&=&3abc \\ a^2 &=& 2(b+c) \\ \end{aligned}$

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.