The sequence $a_{1}, a_{2}, a_{3}, \cdots$ satisfies $a_{1}=\frac{1}{2}$, and $a_{1}+a_{2}+\cdots+a_{n}=n^{2} a_{n}(n \geqslant$ $1)$, determine the value of $a_{n}(n \geqslant 1)$.
Let $S_{n}=a_{1}+a_{2}+\cdots+a_{n}$, then from $a_{n}=S_{n}-S_{n-1}=n^{2} a_{n}-(n-1)^{2} a_{n-1}$, we get $a_{n}=\frac{n-1}{n+1} a_{n-1}, n \geqslant 2$. Therefore,
$$
a_{n}=\frac{n-1}{n+1} a_{n-1}=\frac{(n-1)(n-2)}{(n+1) n} a_{n-2}=\cdots=\frac{(n-1)!}{(n+1) n \cdots 3} a_{1}=\frac{1}{n(n+1)} .
$$