
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
Isosceles trapezoid  $ABCD$  has side lengths  $AB = 6$  and  $CD = 12$ , while  $AD = BC$ . It is given that  $O$ , the circumcenter of  $ABCD$ , lies in the interior of the trapezoid. The extensions of lines  $AD$  and  $BC$  intersect at  $T$ . Given that  $OT = 18$ , the area of  $ABCD$  can be expressed as  $a + b\sqrt{c}$  where  $a$ ,  $b$ , and  $c$  are positive integers where  $c$  is not divisible by the square of any prime. Compute  $a+b+c$ .

*Proposed by Andrew Wen*

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