
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
Consider all triangles \( ABC \) adhering to the conditions: \( AB = AC \), \( D \) is a point on \( \overline{AC} \) where \( \overline{BD} \perp \overline{AC} \), \( E \) is a point on \( \overline{BC} \) such that \( \overline{DE} \perp \overline{BC} \), and \( AC \) and \( CD \) are integers. Given \( BD^2 = 64 \) and \( DE^2 = 16 \), find the smallest possible perimeter of triangle \( ABC \).
A) \( 20 \)
B) \( 20 + 2\sqrt{5} \)
C) \( 20 + 4\sqrt{5} \)
D) \( 24 \)
E) \( 25 \)

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