
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 $\triangle{XYZ}$, $XY=15$, $\angle{X}=45^\circ$, and $\angle{Z}=30^\circ$. Let $G$, $E$, and $L$ be points on the line $YZ$ such that $XG \perp YZ$, $\angle{XYE} = \angle{EYX}$, and $YL = LZ$. Point $Q$ is the midpoint of segment $GL$, and point $R$ is on ray $XE$ such that $RQ \perp YZ$. Find the square of the length of $XR$ in simplest fractional form, and express your answer as a sum of its numerator and denominator.

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