
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  $P$  lies on the diagonal  $AC$  of square  $ABCD$  with  $AP>CP$ . Let  $O_1$  and  $O_2$  be the circumcenters of triangles  $ABP$  and  $CDP$  respectively. Given that  $AB=12$  and  $\angle O_1 P O_2 = 120^\circ$ , then  $AP=\sqrt{a}+\sqrt{b}$  where  $a$  and  $b$  are positive integers. Find  $a+b$ .

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