
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
In parallelogram $ADEH$, points $B$ and $C$ trisect $\overline{AD}$, and points $G$ and $F$ trisect $\overline{HE}$. The lengths are given as $AD = 6$ and $HE = 6$, while $BH = 2\sqrt{2}$. Determine the area of quadrilateral $WXYZ$ formed inside where lines $AF$, $BH$, $CG$, and $DE$ intersect.

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