
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  $ABCD$  of side length  $10$ . Let  $O_1$ ,  $O_2$ ,  $O_3$ ,  $O_4$  be the circumcenters of  $P AB$ ,  $P BC$ ,  $P CD$ , and  $P DA$ , respectively. Given that  $P A+P B +P C +P D = 23\sqrt2$  and the area of  $O_1O_2O_3O_4$  is  $50$ , the second largest of the lengths  $O_1O_2$ ,  $O_2O_3$ ,  $O_3O_4$ ,  $O_4O_1$  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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