A fractal is constructed by starting with an equilateral triangle of area 1. In each step, we replace each line segment of the boundary with a generator of 4 segments, each of length 1/3 of the original. The area of the resulting Koch snowflake at step $n$ is modeled by $A_n = A_{n-1} + \frac{1}{3} \left(\frac{4}{9}\right)^{n-1}$ for $n \geq 1$ with $A_0 = 1$. Find the limit of the area $A_n$ as $n \to \infty$.