
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  $A C$  and  $C E$  be perpendicular line segments, each of length  $18 .$  Suppose  $B$  and  $D$  are the midpoints of  $A C$  and  $C E$  respectively. If  $F$  be the point of intersection of  $E B$  and  $A D,$  then the area of  $\triangle B D F$  is?

[list=1]
[*]  $27\sqrt{2}$ [*]  $18\sqrt{2}$ [*] 13.5
[*] 18
[/list]

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