
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 isosceles triangle  $ABC$  sides  $AB$  and  $BC$  have length  $125$  while side  $AC$  has length  $150$ . Point  $D$  is the midpoint of side  $AC$ .  $E$  is on side  $BC$  so that  $BC$  and  $DE$  are perpendicular. Similarly,  $F$  is on side  $AB$  so that  $AB$  and  $DF$  are perpendicular. Find the area of triangle  $DEF$ .

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