
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 $\triangle ABC$, $\angle BAC=60^{\circ}$ and $\angle ACB=30^{\circ}$. $AB=2$. Let $N$ be the midpoint of segment $\overline{AB}$. Point $D$ lies on side $\overline{BC}$ such that $\overline{AD}\perp\overline{BN}$. Extend segment $\overline{BC}$ through $C$ to point $E$ such that $DE=EB$. Then $CE=\frac{p-\sqrt{q}}{r}$, where $p$ and $r$ are relatively prime positive integers, and $q$ is a positive integer. Find $p+q+r$.

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