Distribution of $U + Y$ $\mod 1$

Question

Let $U$ ~ Unif$[0,1]$ be uniformly distributed on $[0,1]$. Let $Y$ be some random variable, independent from $U$. What is the distribution of the random variable defined as $X=U+Y \mod 1$?

Can someone open my eyes please and tell me how to approach such a problem? I have tried conditioning on $U$, but I didn't get to the answer.

Answer 1

Hint: Condition on $Y$. The distribution of $X$ given $Y=y$ is quite simple.