
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
The random variable \( X \) is defined by the distribution function

$$
F(x)= \begin{cases}
0 & \text{if } x \leq 0 \\
\frac{(1-\cos x)}{2} & \text{if } 0 < x \leq \pi \\
1 & \text{if } x > \pi
\end{cases}
$$

Find the probability density function of the variable \( X \). Calculate the probability that the random variable \( X \) takes values in the interval \( (\pi / 3, \pi / 2) \).

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