
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
Let \(ABTCD\) be a convex pentagon with area 22 such that \(AB = CD\) and the circumcircles of triangles \(TAB\) and \(TCD\) are internally tangent. Given that \(\angle ATD = 90^{\circ}\), \(\angle BTC = 120^{\circ}\), \(BT = 4\), and \(CT = 5\), compute the area of triangle \(TAD\).

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