G3.3 If $f(n)=a^{n}+b^{n}$, where $n$ is a positive integer and $f(3)=[f(1)]^{3}+f(1)$, find the value of $a \cdot b$.
$\begin{array}{l}f(1)=a+b \\ f(3)=(a+b)^{3}+a+b=a^{3}+b^{3} \\ a^{2}+2 a b+b^{2}+1=a^{2}-a b+b^{2} \\ 3 a b=-1 \\ \Rightarrow a b=-\frac{1}{3}\end{array}$