Quadruples of integers with $20^x + 14^{2y} = (x + 2y + z)^{zt}.$

Question

Determine all quadruples $(x,y,z,t)$ of positive integers such that $$20^x + 14^{2y} = (x + 2y + z)^{zt}.$$

We can check that $20+14^2=216=(1+2+3)^3$. But how can we check if there are other ones?