
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 $\triangle ABC$, $\measuredangle BAC = 60^\circ$ and $\measuredangle ACB = 30^\circ$. $AB = 2$. Let $M$ be the midpoint of segment $\overline{AB}$. Point $D$ lies on side $\overline{BC}$ such that $\overline{AD} \perp \overline{CM}$. Extend segment $\overline{BC}$ through $C$ to point $E$ such that $DE = EC$. Find the length of $CE$.

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