If real numbers $a$, $b$ and $c$ satisfy $abc=-1$, $a+b+c=4$, $\frac{a}{a^2-3a-1}+\frac{b}{b^2-3b-1}+\frac{c}{c^2-3c-1}=\frac{4}{9}$, what is the value of $a^2+b^2+c^2$?