
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 lengths of the three sides of a triangle be integers \( l, m, n \) with \( l > m > n \). It is given that \(\left\{\frac{3^{l}}{10^{4}}\right\}= \left\{\frac{3^{m}}{10^{4}}\right\}=\left\{\frac{3^{n}}{10^{4}}\right\}\), where \(\{x\}=x-[x]\) and \([x]\) denotes the greatest integer less than or equal to \(x\). Find the minimum perimeter of such a triangle.

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