We are all related

Algebra Level 2

\[ \begin{array} { l l l l l l l } \large & 2^x & = & 3^y & = & 4^z & = & 576 \\ & \frac 1 x & + & \frac 1 y & + & \frac 1 z & = & ? \\ \end{array} \]

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