4. In the sequence $\left\{a_{n}\right\}$, $a_{1}=1$. When $n \geqslant 2$, $a_{n}, S_{n}, S_{n}-\frac{1}{2}\left(S_{n}\right.$ is the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$) form a geometric progression, then $\lim _{n \rightarrow \infty} n^{2} a_{n}=$ $\qquad$