
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 $Q$ be a point inside triangle $DEF$ such that $Q$ divides each median in the ratio 3:1 when measured from the vertex. Let $H_1$, $H_2$, and $H_3$ be the centroids of triangles $QDE$, $QEF$, and $QFD$, respectively. If the area of triangle $DEF$ is 36, find the area of triangle $H_1 H_2 H_3$.

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