Eccoci.

The following is a function defined for all positive rational values of $x$.

$$ f(x)=\begin{cases} x &x\text{ is integral}\\ f(\frac 1{1-x}) &x \lt 1\\ f\Big(\frac 1{\lceil x\rceil -x}-1+f(x-1)\Big) &\text{otherwise}\end{cases} $$

For example, $f(3/2)=3$, $f(1/6) = 65533$ and $f(13/10) = 7625597484985$.

Find $f(22/7)$. Give your answer modulo $10^{15}$.