
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
In square $WXYZ$, point $P$ lies on diagonal $WY$ such that $WP > PY$. Let $O_3$ and $O_4$ be the circumcenters of triangles $WXP$ and $ZYP$, respectively. Given that $WZ = 10$ and $\angle O_{3}PO_{4} = 90^{\circ}$, find $WP$ in the form $\sqrt{c} + \sqrt{d}$ where $c$ and $d$ are positive integers, and calculate $c + d$.

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