
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 \( AP: PB = 2:1 \) and \( AQ: QC = 1:3 \). Point \( M \) is chosen on side \( BC \) completely at random. Find the probability that the area of triangle \( ABC \) does not exceed the area of triangle \( PQM \) by more than three times. Find the expected value of the random variable - the ratio of the areas of triangles \( PQM \) and \( ABC \).

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