
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 \(PA \cdot PB = 2\), \(PC \cdot PD = 18\), and \(PB \cdot PC = 9\). 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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