Let $k$ be the $x$-coordinate of the intersection point of the curve $y = \left(\frac{1}{5}\right)^{x-3}$ and the line $y = x$. A function $f(x)$ defined on the set of all real numbers satisfies the following conditions.\\
For all real numbers $x > k$, $f(x) = \left(\frac{1}{5}\right)^{x-3}$ and $f(f(x)) = 3x$.\\
What is the value of $f\left(\frac{1}{k^{3} \times 5^{3k}}\right)$? [4 points]