Why exponents of variables?!

Algebra Level pending

Let \((x_{1},y_{1}),(x_{2},y_{2}),\dots,(x_{n},y_{n})\) be the integer solutions of the system of equations.

\[ \begin{cases} x^{x+y} = y^{12} \\ y^{x+y} = x^{3} \\ \end{cases}\]

Find the value of \(\displaystyle \sum\limits_{i=1}^{n} (x_{i}+y_{i})\).

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