BALTIC-WAY 2022 Problem 19

Find all triples $(x, y, z)$ of nonnegative integers such that $x^5+x^4+1=3^y7^z$

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.