# PDF of conditional uniformly distributed random variable

Given two barrels of water, $$A,B$$, with 1 liter each. We pour an $$X\sim U[0,1]$$ amount of water from $$A$$ to $$B$$ and then $$Y$$ amount of water randomly from $$B$$ to $$A$$ $$(Y|X=x\sim U[0,1+x])$$.

Calculate the CDF of $$Z$$ - the amount of water in barrel A after both transfers.

My try:

Obviously $$Z=1-X+Y$$, so for $$t\in[0,2]$$, we calculate (using law of total probability):

$$F_Z(t)=\mathbb{P}(Z\le t)=\mathbb{P}(1-X+Y\le t)=\int_0^1\mathbb{P}(Y\le t+x-1|X=x)f_X(x)=\int_0^1\mathbb{P}(Y\le t+x-1|X=x)$$

How can I determine the limits of integration given different values of $$t$$?

Any help would be appreciated