
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 $\triangle XYZ$, $\angle XYZ = 60^\circ$ and $\angle YXZ = 30^\circ$. Let $XY = 1$. Let $N$ be the midpoint of segment $\overline{XY}$. Point $D$ lies on side $\overline{XZ}$ such that $\overline{YD} \perp \overline{ZN}$. Extend segment $\overline{YZ}$ through $Y$ to point $B$ such that $DB = DY$. Find $YB$ in the form $\frac{p - \sqrt{q}}{r}$, where $p$ and $r$ are relatively prime positive integers, and $q$ is a positive integer. Then determine $p + q + r$.

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