
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 $EFGH$, point $Q$ lies on the diagonal $EH$ such that $EQ > HQ$. Let $R_1$ and $R_2$ be the circumcenters of triangles $EFQ$ and $GHQ$ respectively. Given that $EF = 10$ and $\angle R_{1}QR_{2} = 150^{\circ}$, find the value of $EQ$ if it can be expressed as $\sqrt{x} + \sqrt{y}$ where $x$ and $y$ are positive integers.

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