
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]


system
Independent random variables \( X \) and \( Y \) are given by the following probability density functions:

\[
f_{1}(x) = e^{-x} \quad (0 \leq x < \infty), \quad f_{2}(y) = \frac{1}{2} e^{-y / 2} \quad (0 \leq y < \infty).
\]

Find the density function of the random variable \( Z = X + Y \).

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