10. (20 points) Let the ellipse $\Gamma: \frac{x^{2}}{a^{2}}+y^{2}=1(a>1)$, any circle with center $A(0,1)$ intersects the ellipse $\Gamma$ at most at three points. Find the range of the eccentricity of the ellipse.

 untranslated part:
设椭圆 $\Gamma: \frac{x^{2}}{a^{2}}+y^{2}=1(a>1)$,任意以 $A(0,1)$ 为圆心的圆与椭圆 $\Gamma$ 至多有三个公共点. 求椭圆离心率的取值范围.

 translated part:
Let the ellipse $\Gamma: \frac{x^{2}}{a^{2}}+y^{2}=1(a>1)$, any circle with center $A(0,1)$ intersects the ellipse $\Gamma$ at most at three points. Find the range of the eccentricity of the ellipse.
10. Let the line $l: y=k x+1(k>0)$.

Combining with the ellipse equation, we get
$$
\left(\frac{1}{a^{2}}+k^{2}\right) x^{2}+2 k x=0 .
$$

Let $A$ be one of the intersection points of $l$ and the ellipse $\Gamma$, and let the other intersection point be $B$. Thus,
$$
\left|x_{A}-x_{B}\right|=\frac{2 k a^{2}}{1+a^{2} k^{2}} \text {. }
$$

Then $|A B|=\sqrt{1+k^{2}} \frac{2 k a^{2}}{1+a^{2} k^{2}}$.
Let $x=k^{2}$.
Consider the function $f(x)=\frac{x(1+x)}{\left(1+a^{2} x\right)^{2}}$.
It is easy to see that, $f^{\prime}(x)=\frac{\left(2-a^{2}\right) x+1}{\left(1+a^{2} x\right)^{3}}$.
When $a^{2} \leqslant 2$, i.e., $a \leqslant \sqrt{2}$, $f(x)$ is monotonically increasing, and $\sqrt{f(x)}$ is monotonically increasing;
When $a^{2}>2$, there exist positive real numbers $x_{1} \neq x_{2}$, such that
$$
f\left(x_{1}\right)=f\left(x_{2}\right) \Rightarrow \sqrt{f\left(x_{1}\right)}=\sqrt{f\left(x_{2}\right)} \text {. }
$$

The eccentricity $e=\frac{a^{2}-1}{a^{2}}=1-\frac{1}{a^{2}} \in\left(0, \frac{1}{2}\right]$.