
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 area $36$. Point $E$ lies on side $\overline{BC}$. Points $F$ and $H$ lie on $\overline{AE}$ with $AF:FE:EH = 1:2:2$. Point $G$ is the midpoint of $\overline{DE}$. Given that quadrilateral $BEGF$ has area $5$, what is the area of triangle $GCD$?

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