OPM 2016 Problem 5

Determine all natural numbers $x, y$ and $z$ such that the number $2^x +4^y +8^z +16^2$ is a power of $2$ .

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.