
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
A point $Q$ is chosen in the interior of $\triangle DEF$ such that when lines are drawn through $Q$ parallel to the sides of $\triangle DEF$, the resulting smaller triangles $u_1$, $u_2$, and $u_3$ have areas $16$, $25$, and $36$, respectively. Additionally, line segments connecting $Q$ with each vertex of $\triangle DEF$ are bisectors of the angles at vertices $D$, $E$, $F$. Calculate the area of $\triangle DEF$.

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