4. Given the curve $C_{1}: y=\sqrt{-x^{2}+10 x-9}$ and point $A(1,0)$. If there exist two distinct points $B$ and $C$ on curve $C_{1}$, whose distances to the line $l: 3 x+1=0$ are $|A B|$ and $|A C|$, respectively, then $|A B|+|A C|=$
4.8.

Let points $B\left(x_{B}, y_{B}\right), C\left(x_{C}, y_{C}\right)$, the locus of points whose distance to point $A(1,0)$ is equal to the distance to the line $l: x=-\frac{1}{3}$ is given by the equation $y^{2}=\frac{8}{3} x-\frac{8}{9}$.
Solving the system of equations
$$
\begin{array}{l}
\left\{\begin{array}{l}
y^{2}=\frac{8}{3} x-\frac{8}{9}, \\
x^{2}+y^{2}-10 x+9=0
\end{array}\right. \\
\Rightarrow x^{2}-\frac{22}{3} x+\frac{73}{9}=0 \\
\Rightarrow|A B|+|A C|=x_{B}+x_{C}+\frac{2}{3}=8 \text {. } \\
\end{array}
$$