Equality of continued fraction expansions

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Consider the two continued fraction expansions

$$ F(q,z) = \frac{1}{1- \frac{z}{1-qzF(q,qz)}} $$

and

$$ G(q,z) = \frac{q^{-1}}{1 - \frac{qz}{1-qzG(q,qz)}}. $$

Is it true that $F(q,z) = G(q,z)$?