probability - Find CDF of random variable which depends on other variable

Let $X$ be a random variable with uniform distribution on $[-1, 1]$. Find the CDF of random variable Y given by the following formula:

$Y = \left\{\begin{matrix}
-\frac{1}{2},& X < - \frac{1}{2}\\
X,& -\frac{1}{2} \leq X \leq \frac{1}{4}\\
\frac{1}{4}, & X > \frac{1}{4}
\end{matrix}\right.$

So I've found PDF and CDF of $X$:

$f_X(x) = \begin{cases}

\frac{1}{2}, & x \in [-1, 1]\\
0, & \text{otherwise}
\end{cases}$

$F_X(a) = \int_{-\infty}^{a} f_X(x) dx = \left\{\begin{matrix}
0, & a \leq -1\\
\frac{a+1}{2}, & a \in (-1, 1) \\
1, & a \geq 1
\end{matrix}\right.$

I tried to find Y's CDF by:

$F_Y(a) = P(Y \leq a)$

$= P(-\frac{1}{2} \leq a, X < - \frac{1}{2}) + P(X \leq a, - \frac{1}{2} \leq X \leq \frac{1}{4}) + P(\frac{1}{4} \leq a, X > -\frac{1}{4})$

But what should I do next? I'm finding such CDF for the first time and my notes say I need to consider a few different cases, but I have no clue what they should look like and how to do it. Any tips would be helpful.