
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 rectangle $ABCD$ with $AP > CP$. Let $O_1$ and $O_2$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. The side lengths of the rectangle are $AB = 15$ and $BC = 8$. If $\angle O_1PO_2 = 90^{\circ}$, express $AP$ in the form $\sqrt{a} + \sqrt{b}$, where $a$ and $b$ are positive integers, and find $a + b$.

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