
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 $EFGH$ with $EQ > GQ$. Let $R_{1}$ and $R_{2}$ be the circumcenters of triangles $EFQ$ and $GHQ$ respectively. Given that $EF = 8$ and $\angle R_{1}QR_{2} = 90^{\circ}$, find the length $EQ$ in the form $\sqrt{c} + \sqrt{d}$, where $c$ and $d$ are positive integers. Find $c+d$.

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