A system of sums

Algebra Level 5

\[\begin{cases} 1 + x + xy + x^2y + x^2y^2 + x^3y^2 + x^3y^3 +\ldots = \frac {21}{5} \\ 1 + y + xy + xy^2 + x^2y^2 + x^2y^3 + x^3y^3 + \ldots = 4 \end{cases}\]

Suppose there are two numbers \(x\) and \(y\) such that they satisfy the system of equations above. Find the value of \(z \) such that

\[1 + \frac{x}{2} + \frac{xy}{4} + \frac{xyz}{8} + \frac {x^2yz}{16} + \frac{x^2y^2z}{32} + \frac{x^2y^2z^2}{64} +\ldots = \frac {216}{133}. \]

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