
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 \( EFGH \) has a side length of \( 40 \). Point \( Q \) lies inside the square such that \( EQ = 16 \) and \( FQ = 34 \). The centroids of triangles \( \triangle{EFQ} \), \( \triangle{FGQ} \), \( \triangle{GHQ} \), and \( \triangle{HEQ} \) are the vertices of a convex quadrilateral. What is the area of this quadrilateral?

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