
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. Given that $AB = 8$ and $CD = 15$, and $\angle O_{1}PO_{2} = 90^{\circ}$, express $AP$ in terms of $a$ and $b$ where $\sqrt{a} + \sqrt{b}$ are positive integers. Find $a + b$.

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