
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
Let the three sides of a triangle be integers \( l \), \( m \), and \( n \) with \( l > m > n \). It is known that \( \left\{\frac{3^l}{10^4}\right\} = \left\{\frac{3^m}{10^4}\right\} = \left\{\frac{3^n}{10^4}\right\} \), where \( \{x\} \) denotes the fractional part of \( x \). Determine the minimum value of the perimeter of the triangle.

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