
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 \(ABCD\) be a trapezoid with \(AB \parallel CD\), \(AB = 13\), \(BC = 8\), \(CD = 23\), and \(DA = 10\). Bisectors of \(\angle A\) and \(\angle D\) meet at \(P\), and bisectors of \(\angle B\) and \(\angle C\) meet at \(Q\). Determine the area of the hexagon \(ABQCDP\).
**A** \(25\sqrt{3}\)
**B** \(30\sqrt{3}\)
**C** \(34\sqrt{3}\)
**D** \(36\sqrt{3}\)
**E** \(40\sqrt{3}\)

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