
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 parallelogram ABCD, P is the midpoint of BC. A line through P parallel to BD intersects CD at Q. Connecting PA, PD, QA, and QB, calculate the number of triangles in the diagram that have the same area as triangle ABP, excluding triangle ABP itself.

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