
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  $ABCD$  be a parallelogram. Let  $E$  and  $F$  be the midpoints of sides  $AB$  and  $BC$  respectively. The lines  $EC$  and  $FD$  intersect at  $P$  and form four triangles  $APB, BPC, CPD, DPA$ . If the area of the parallelogram is  $100$ , what is the maximum area of a triangles among these four triangles?

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