Example 8 Given that $a$, $b$, and $c$ are all positive integers, and the parabola $y=a x^{2}+b x+c$ intersects the $x$-axis at two distinct points $A$ and $B$. If the distances from $A$ and $B$ to the origin are both less than 1, find the minimum value of $a+b+c$.
$(1996$, National Junior High School Mathematics League)
$$
\begin{array}{l}
\text{Analysis: Let } x_{1} \text{ and } x_{2} \text{ be the x-coordinates of the points where the parabola intersects the x-axis, then } x_{1}+x_{2}=-\frac{b}{a}, x_{1} x_{2}=\frac{c}{a}. \text{ It is easy to know that } -\frac{b}{a}<0, \text{ and } -116 c(a-b+c). \\
\because a-b+c>0, c \geqslant 1, \\
\therefore a^{2}>16, a>4 .
\end{array}
$$
$>20 c$. Thus, the minimum value of $b$ is $5$, the minimum value of $c$ is $1$, and the parabola is $y=5 x^{2}+5 x+1$, intersecting the x-axis at $\left(\frac{-5+\sqrt{5}}{10}, 0\right),\left(\frac{-5-\sqrt{5}}{10}, 0\right)$, which meets the requirements.
Therefore, $(a+b+c)_{\text{min}}=11$.