Victor's Level 5 Challenges - 2

Let the three sides of a triangle be integers \(l,m,n\) with \(l>m>n\) and satisfy the equation

\[ \large \left\{\frac{3^l}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}\]

Find the minimum perimeter of such a triangle.

Details and Assumptions:

\(\{x\}=x-\left[x\right]\) and \(\left[x\right]\) denotes the integral part of the number \(x\).

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