
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 \(ABC\), it is known that \(AB = BC\) and \(AC = 4 \sqrt{3}\). The radius of the inscribed circle is 3. Line \(AE\) intersects the altitude \(BD\) at point \(E\), and the inscribed circle at points \(M\) and \(N\) (with \(M\) being between \(A\) and \(E\)). Given that \(ED = 2\), find \(EN\).

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