
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  $ABC$  be a triangle with  $\angle BAC = 90^\circ$ . Construct the square  $BDEC$  such as  $A$  and the square are at opposite sides of  $BC$ . Let the angle bisector of  $\angle BAC$  cut the sides  $[BC]$  and  $[DE]$  at  $F$  and  $G$ , respectively. If  $|AB|=24$  and  $|AC|=10$ , calculate the area of quadrilateral  $BDGF$ .

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