
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  $\ell, m, n$ , respectively, satisfying  $\ell>m>n$  and  $\left\{\frac{3^\ell}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}$ , where  $\{x\}=x-\lfloor{x}\rfloor$  and  $\lfloor{x}\rfloor$  denotes the integral part of the number  $x$ . Find the minimum perimeter of such a triangle.

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