
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


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Given the probability densities of two independent uniformly distributed random variables $X$ and $Y$:

$f_{1}(x)=\frac{1}{2}$ in the interval $(0, 2)$, outside this interval $f_{1}(x)=0$

$f_{2}(y)=\frac{1}{2}$ in the interval $(0, 2)$, outside this interval $f_{2}(y)=0$

Find the cumulative distribution function (CDF) and probability density function (PDF) of the random variable $Z = X + Y$. Plot the density function $g(z)$.

Present the answer in LaTex format: \boxed{Your answer}
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