Chemiluminescent Cubic

\[ \large \begin{cases} { 4x }^{ 3 }+3{ x }^{ 2 }y+{ y }^{ 3 }=8 \\ { 2x }^{ 3 }-2{ x }^{ 2 }y+x{ y }^{ 2 }=1 \end{cases}\]

Let \(\left( { x }_{ 1 },{ y }_{ 1 } \right), \ldots ,\left( { x }_{ n },{ y }_{n} \right) \) be integers satisfying above equation.

Evaluate \[\large \sum_{i = 1}^{n} x_{i}^{ 6 } - y_{i}^{6} \]

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