
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 $Q$ lies on the diagonal $AC$ of square $WXYZ$ such that $WQ > QZ$. Let $O_{3}$ and $O_{4}$ be the circumcenters of triangles $WZQ$ and $XYZ$ respectively. If $WZ = 10$ and $\angle O_{3}QO_{4} = 90^{\circ}$, calculate the length of $QZ = \sqrt{m} - \sqrt{n}$, where $m$ and $n$ are integers. Find $m+n$.

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