
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
Square $ABCD$ has a side length of $40$. Point $Q$ lies within the square such that $AQ = 16$ and $BQ = 34$. Determine the area of the quadrilateral formed by the centroids of $\triangle{ABQ}$, $\triangle{BCQ}$, $\triangle{CDQ}$, and $\triangle{DAQ}$ within the square.

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