
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
In the study of income or wealth distribution, in some cases, each specific value of wealth can be considered as the result of the action of a large number of mutually independent factors, each of which causes some increment proportional to the already achieved level of income. In this case, the size of income can be considered distributed according to the so-called log-normal law, where the probability density is defined by the formula

$$
p(x)=\left\{\begin{array}{cc}
0 & \text { for } x \leqslant 0 \\
\frac{1}{\sigma x \sqrt{2 \pi}} \cdot e^{-\frac{(\ln x-a)^{2}}{2 \sigma^{2}}} & \text { for } x > 0
\end{array}\right.
$$

Find the expected value and variance of this random variable.

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