
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
Given an isosceles triangle \( ABC \) with \( |AB| = |BC| \). \( AD \) is the angle bisector. A perpendicular erected to \( AD \) at point \( D \) intersects the extension of \( AC \) at point \( E \). The feet of the perpendiculars dropped from \( B \) and \( D \) to \( AC \) are \( M \) and \( N \) respectively. Find \( |MN| \) if \( |AE| = a \).

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