
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
According to the problem conditions, the random variable \(\xi = \frac{CM}{CB}\) is uniformly distributed over the interval \([0, 1]\). If \(x\) is the value of the random variable \(\xi\), then

$$
\mathrm{s}(x) = \mathrm{S}_{PQM} = S_{ABC} \cdot \left(1 - \frac{1}{6} - \frac{3x}{4} - \frac{(1-x)}{3}\right) = \frac{S_{ABC}}{12} (6-5x) \rightarrow \frac{\mathrm{S}_{PQM}}{S_{ABC}} \geq \frac{1}{3}
$$

Then the condition \(\frac{S_{PQM}}{S_{ABC}} \geq \frac{1}{3}\) can be written as

$$
\frac{6-5x}{12} \geq \frac{1}{3}
$$

From which we find \(x \leq \frac{2}{5}\). Given the uniform distribution, \(P(A) = \frac{2}{5}\).

Find the expected value of the random variable \(X = \frac{S_{PQM}}{S_{ABC}}\)

$$
M_{X} = \int_{0}^{1} \frac{6-5x}{12} \, dx = -\left. \frac{1}{120}(6-5x)^{2} \right|_{0}^{1} = \frac{7}{24}
$$

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