
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
Let \( \triangle ABC \) have centroid \( S \), midpoint of segment \( AS \) be \( H \), and midpoint of side \( AB \) be \( Q \). Let the line parallel to \( BC \) through \( H \) intersect \( AB \) at \( P \) and line \( CQ \) at \( R \). What is the ratio of the areas of triangles \( PQR \) and \( APH \)?

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