HMMT 2006 U35

A sequence is defined by $A_{0}=0, A_{1}=1, A_{2}=2$, and, for integers $n \geq 3$,

$$A_{n}=\frac{A_{n-1}+A_{n-2}+A_{n-3}}{3}+\frac{1}{n^{4}-n^{2}}$$

Compute $\lim _{N \rightarrow \infty} A_{N}$.