Sum of Distribution Functions

Question

Given $X$ follows distribution function $F$, $Y$ follows $G$ and $Z$ follows $H$
Express $Z=f(X,Y)$. Say $H=\lambda F+(1-\lambda)G\,\,\,\,(\lambda\in[0,1])$
Then find $f$

I was able to verify that $H$ is a valid df but I am unable to find $f$. Any hints?

Answer 1

I was able to find the answer. Define $T=Bern(\lambda)$
Define $Z=T(X-Y)+Y$
Then distribution function for $Z$ is $H$