
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

A random variable \( X \) is given by its cumulative distribution function

$$
F(x)=\left\{
\begin{array}{ll}
0 & \text{when } x \leqslant 1 \\
a \ln x & \text{when } 1 < x \leqslant e \\
1 & \text{when } x > e
\end{array}
\right.
$$

Required: 
a) Find the value of parameter \( a \);
b) Find the probability density function \( f(x) \);
c) Plot the functions \( y=F(x) \) and \( y=f(x) \);
d) Find the probability that \( X \) belongs to the interval \( \left(\frac{1}{\sqrt[3]{e}}, \sqrt{e}\right) \).

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