
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
Points $P$ and $Q$ are located on the sides $AB$ and $AC$ of triangle $ABC$ such that $AP : PB = 1 : 3$ and $AQ : QC = 1 : 1$. Point $M$ is chosen randomly on the side $BC$. Find the probability that the area of triangle $ABC$ exceeds the area of triangle $PQM$ by no more than six times. Find the expected value of the random variable representing the ratio of the areas of triangles $PQM$ and $ABC$.

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