
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
Given triangle $ABC$ with $|AB|=18$, $|AC|=24$, and $m(\widehat{BAC}) = 150^\circ$. Points $D$, $E$, $F$ lie on sides $[AB]$, $[AC]$, $[BC]$ respectively, such that $|BD|=6$, $|CE|=8$, and $|CF|=2|BF|$. Find the area of triangle $H_1H_2H_3$, where $H_1$, $H_2$, $H_3$ are the reflections of the orthocenter of triangle $ABC$ over the points $D$, $E$, $F$.

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