
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
Let $P$ be a point on diagonal $AC$ of square $ABCD$ such that $AP = 3CP$. The side length of square $ABCD$ is 10 units. Let $O_{1}$ and $O_{2}$ be the circumcenters of triangles $ABP$ and $CDP$, respectively. If $\angle O_{1}PO_{2} = 90^{\circ}$, find the lengths of $AP$ and $CP$ in terms of square roots.

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