
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
Given a quadrilateral \(A B C D\), see the image. Point \(T_{1}\) is the centroid of triangle \(B C D\), point \(T_{2}\) is the centroid of triangle \(A B D\), and points \(T_{1}\) and \(T_{2}\) lie on segment \(A C\). The length of segment \(T_{1} T_{2}\) is \(3 \text{ cm}\) and point \(D\) has a distance of \(3 \text{ cm}\) from segment \(A C\).

Determine the area of quadrilateral \(A B C D\).

(E. Patáková)

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