
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

Point \(P\) is located inside a square \(ABCD\) with side length 10. Let \(O_1, O_2, O_3\), and \(O_4\) be the circumcenters of \(\triangle PAB, \triangle PBC, \triangle PCD\), and \(\triangle PDA\), respectively. Given that \(PA + PB + PC + PD = 23\sqrt{2}\) and the area of quadrilateral \(O_1O_2O_3O_4\) is 50, the second largest of the lengths \(O_1O_2, O_2O_3, O_3O_4,\) and \(O_4O_1\) can be written as \(\sqrt{\frac{a}{b}}\), where \(a\) and \(b\) are relatively prime positive integers. Compute \(100a + b\).

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