
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
On one side of the acute angle \(A\), points \(P\) and \(Q\) are marked such that \(AP = 4\), \(AQ = 12\). On the other side, points \(M\) and \(N\) are marked at distances of 6 and 10 from the vertex. Find the ratio of the areas of triangles \(MNO\) and \(PQO\), where \(O\) is the intersection point of the lines \(MQ\) and \(NP\).

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