
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
In triangle \(ABC\), points \(M\), \(N\), and \(P\) are chosen such that \(M\) and \(N\) lie on sides \(AC\) and \(BC\) respectively, and \(P\) lies on segment \(MN\). The following conditions are satisfied:

\[
\frac{|AM|}{|MC|} = \frac{|CN|}{|NB|} = \frac{|MP|}{|PN|}
\]

Find the area of triangle \(ABC\) given that the areas of triangles \(AMP\) and \(BNP\) are \(T\) and \(Q\) respectively.

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