11. (20 points) Let $\triangle A B C$ be an inscribed triangle in the ellipse $\Gamma: \frac{x^{2}}{4}+y^{2}=1$, where $A$ is the intersection of the ellipse $\Gamma$ with the positive $x$-axis, and the product of the slopes of lines $A B$ and $A C$ is $-\frac{1}{4}$. Let $G$ be the centroid of $\triangle A B C$. Find the range of $|G A|+|G B|+|G C|$.
11. It is known that the line $BC$ passes through the origin.

At this point, $G\left(\frac{2}{3}, 0\right), |GA| = \frac{3}{4}$ (a constant).
Let the points be $B\left(x_{1}, y_{1}\right), C\left(x_{2}, y_{2}\right)$.
Finding the range of $|GB| + |GC|$ is equivalent to finding the range of the function
$$
f(x, y) = \sqrt{\left(x - \frac{2}{3}\right)^{2} + y^{2}} + \sqrt{\left(x + \frac{2}{3}\right)^{2} + y^{2}}
$$

where $\frac{x^{2}}{4} + y^{2} = 1$.
Let $G\left(\frac{2}{3}, 0\right), G_{1}\left(-\frac{2}{3}, 0\right)$,
$$
2a = \sqrt{\left(x - \frac{2}{3}\right)^{2} + y^{2}} + \sqrt{\left(x + \frac{2}{3}\right)^{2} + y^{2}}.
$$

The curve corresponding to equation (1) is an ellipse with foci at $G$ and $G_{1}$ and a major axis length of $2a$.
(1) When equation (1) passes through the points $(0,1)$ and $(0,-1)$, by the definition of an ellipse, equation (1) can be transformed into $\frac{x^{2}}{\frac{13}{9}} + y^{2} = 1$.

At this point, any point $P\left(x_{0}, y_{0}\right)$ on the ellipse $\frac{x^{2}}{4} + y^{2} = 1$ satisfies
$$
\frac{x_{0}^{2}}{\frac{13}{9}} + y_{0}^{2} \geq \frac{x_{0}^{2}}{4} + y_{0}^{2} = 1,
$$

with equality holding if and only if $x = 0, y = \pm 1$. Thus, all other points on the ellipse $\Gamma$ lie outside the ellipse $\frac{x^{2}}{\frac{13}{9}} + y^{2} = 1$.

Therefore, when equation (1) passes through other points on the ellipse $\Gamma$, $2a$ increases.
Thus, $(2a)_{\min} = \frac{2\sqrt{13}}{3}$.
(2) When equation (1) passes through the points $(2,0)$ and $(-2,0)$, by the definition of an ellipse, equation (1) can be transformed into $\frac{x^{2}}{4} + \frac{y^{2}}{\frac{32}{9}} = 1$.

At this point, any point $P\left(x_{0}, y_{0}\right)$ on the ellipse $\frac{x^{2}}{4} + y^{2} = 1$ satisfies
$$
\frac{x_{0}^{2}}{4} + \frac{y_{0}^{2}}{\frac{32}{9}} \leq \frac{x_{0}^{2}}{4} + y_{0}^{2} = 1,
$$

with equality holding if and only if $x = \pm 2, y = 0$. Thus, all other points on the ellipse $\Gamma$ lie inside the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{\frac{32}{9}} = 1$.

Therefore, when equation (1) passes through other points on the ellipse $\Gamma$, $2a$ decreases.
Thus, $2a \leq 4$.
However, equality holds when one of $B$ or $C$ coincides with $A$, so the equality does not hold.

In summary, when equation (1) passes through other points on the ellipse $\Gamma$, by the intermediate value theorem, $2a \in \left[\frac{2\sqrt{13}}{3}, 4\right)$.
Thus, the range of $|GA| + |GB| + |GC|$ is
$$
\left[\frac{2\sqrt{13} + 4}{3}, \frac{16}{3}\right).
$$