Given $2^x+3^x=4$, find all non-trivial rational $a, b, c$ such that $a^x+b^x=c$, or prove that none exist.
(Trivial solutions include those with $a$ and $b$ as $0$ or $1$, for example $a=b=c=0$; or $a=b=1, c=2$, etc. The other trivial solution is $a=2, b=3, c=4$.)
I have not found any such $a, b, c$. Are there any? I have tried setting $a=\frac{p}{q}$ etc., as well as raising both sides of $2^x+3^x=4$ to positive integer powers, to no avail.
(This question was inspired by How to find the solution to $4^x+9^x=4$.)