Solving system of congruences involving powers

I am in the middle of a problem which needs showing that the following system of congruences has finite number of solutions. I have verified up to some extent through sage that this has only two solutions for $(q, r)$ (with $q<r$) namely $(11, 17)$ and $(23, 103)$. The system of congruences is the following:

$$ 3^{3q-1} \equiv 1 \pmod r,$$ $$ q^{3q-1} \equiv 1 \pmod r, $$ $$ 3^{3r-1} \equiv 1 \pmod q,$$ $$ r^{3r-1} \equiv 1 \pmod q.$$

I am unable to prove this and before wasting much time on this, I just wanted to know if the question is really answerable or not. Any help in this direction is greatly appreciated.