I want to find the CDF of $Z=\max\left(\frac{X}{Y},\frac{X}{C}\right)$ when $X>Y$ and $X>C$. Here, $X,Y\geq0$ are independent random variables, and $C>0$ is a constant.

I am facing two difficulties:

 1. I could get the product of CDFs within $\max$ if two were
    independent. However, as they have $X$ in common, dependency is
    created. Therefore, I tried to condition on $X$ first and solve.
    However, in that case, $\frac{X}{C}$ becomes a constant, which is
    confusing me. 
 2. I have more complications in applying the conditions of $X>Y$ and
    $X>C$ to the analysis.

Can someone help me write the CDF of $Z$ in terms of the PDFs/CDFs of $X$ and $Y$?