
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 \( \triangle ABC \), \( M \) is the midpoint of side \( AC \), \( D \) is a point on side \( BC \) such that \( AD \) is the angle bisector of \( \angle BAC \), and \( P \) is the point of intersection of \( AD \) and \( BM \). Given that \( AB = 10 \, \text{cm} \), \( AC = 30 \, \text{cm} \), and the area of triangle \( \triangle ABC \) is \( 100 \, \text{cm}^2 \), calculate the area of triangle \( \triangle ABP \).

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