Example 1 If real numbers $a, b, c$ satisfy $a+b+c=1, a^{2}+b^{2}+c^{2}=2, a^{3}+b^{3}+c^{3}=3$, find the values of $a b c$ and $a^{4}+b^{4}+c^{4}$.
Let $a, b, c$ be the roots of the polynomial $f(t)=t^{3}+a_{1} t^{2}+a_{2} t+a_{3}$. Let $T_{k}=a^{k}+b^{k}+c^{k}(k=1,2,3,4)$, then by Newton's formulas, we have
$T_{1}=1=a+b+c=-a_{1}$, so $a_{1}=-1$.
$T_{2}=-a_{1} T_{1}-2 a_{2}=1-2 a_{2}=2$, so $a_{2}=-\frac{1}{2}$.
$T_{3}=-a_{1} T_{2}-a_{2} T_{1}-3 a_{3}=\frac{5}{2}-3 a_{3}$, so $a_{3}=-\frac{1}{6}$.
Thus, $a b c=-a_{3}=\frac{1}{6}$.
Also, $T_{4}=-a_{1} T_{3}-a_{2} T_{2}-a_{3} T_{1}=3+1+\frac{1}{6}=\frac{25}{6}$,
so $a^{4}+b^{4}+c^{4}=\frac{25}{6}$.