Guide:

Use conditional probability, first figure out $P(Y \le y|Z=z)$ and $f_Z(z)$

Now you can compute the following:

\begin{align} P(Y \le y) &= \int_0^1 P(Y \le y|Z=z)f_Z(z)\, dz \end{align}

Guide:

Use conditional probability, first figure out $P(Y \le y|Z=z)$ and $f_Z(z)$

Now you can compute the following:

\begin{align} P(Y \le y) &= \int_y^1 P(Y \le y|Z=z)f_Z(z)\, dz \end{align}