
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 $ABC$, $\angle BAC = 30^\circ$ and $\angle ACB = 60^\circ$. $BC = 1$. Let $M$ be the midpoint of segment $\overline{BC}$. Point $D$ lies on side $\overline{AC}$ such that $\overline{BD} \perp \overline{AM}$. Extend segment $\overline{AC}$ through $C$ to point $E$ such that $DE = EB$. Find $CE$ expressed in terms of simplest radicals.

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