
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
Triangle  $ABC$  has side lengths  $AB=13, BC=14,$  and  $CA=15$ . Let  $\Gamma$  denote the circumcircle of  $\triangle ABC$ . Let  $H$  be the orthocenter of  $\triangle ABC$ . Let  $AH$  intersect  $\Gamma$  at a point  $D$  other than  $A$ . Let  $BH$  intersect  $AC$  at  $F$  and  $\Gamma$  at point  $G$  other than  $B$ . Suppose  $DG$  intersects  $AC$  at  $X$ . Compute the greatest integer less than or equal to the area of quadrilateral  $HDXF$ .

*Proposed by Kenan Hasanaliyev (claserken)*

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