
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 square $ABCD$, points $E$ and $F$ divide $\overline{AB}$ into three segments with ratios $1:2$, and points $G$ and $H$ divide $\overline{CD}$ similarly. Let $R$ and $T$ be points on $\overline{AD}$ and $\overline{BC}$ such that $\overline{RT}$ divides the square into two equal areas. Also, let $S$ be the midpoint of $\overline{AB}$ and $Q$ be the midpoint of $\overline{CD}$. What is the area of quadrilateral $RQST$ if the side length of square $ABCD$ is $3$?

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