
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 rectangle inscribed in circle  $\Gamma$ , and let  $P$  be a point on minor arc  $AB$  of  $\Gamma$ . Suppose that  $P A \cdot P B = 2$ ,  $P C \cdot  P D = 18$ , and  $P B \cdot  P C = 9$ . The area of rectangle  $ABCD$  can be expressed as  $\frac{a\sqrt{b}}{c}$  , where  $a$  and  $c$  are relatively prime positive integers and  $b$  is a squarefree positive integer. Compute  $100a + 10b + c$ .

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