
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 $\triangle{XYZ}, XY=12, \angle{X}=45^\circ$, and $\angle{Y}=60^\circ$. Let $G, E,$ and $L$ be points on the line $YZ$ such that $XG\perp{YZ}$, $\angle{XYE}=\angle{EYX}$, and $YL=LY$. Point $O$ is the midpoint of the segment $GL$, and point $Q$ is on ray $XE$ such that $QO\perp{YZ}$. Then $XQ^2=\dfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

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