Solve the equation $1+2^x+2^{x+y}=(1+2^n)^z$

Question

For fixed integer $n$ solve the equation $1+2^x+2^{x+y}=(1+2^n)^z$ for integers $x,y,z.$.

It is easy to find two solutions: $z=1,x=n-1,y=0$ and $z=2,x=n+1,y=n-1.$ How to prove that there are not any solution for $z>2?$