
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 $BD$ of square $ABCD$ with $BP > DP$. Let $O_{1}$ and $O_{2}$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB = 8$ and $\angle O_{1}PO_{2} = 135^\circ$, determine $BP$ in the form of $\sqrt{c}$, where $c$ is a positive integer.

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