13. (25 points) The equation of line $m$ is $y=k x+1, A$ and $B$ are two points on line $m$, whose x-coordinates are exactly the two different negative real roots of the quadratic equation in $x$
$$
\left(1-k^{2}\right) x^{2}-2 k x-2=0
$$

The line $l$ passes through point $P(-2,0)$ and the midpoint of segment $A B$, $C D$ is a moving line segment on the $y$-axis. Considering all possible lines $l$, when $l$ and segment $C D$ have no common points, does the maximum length of $C D$ exist? If it exists, find the maximum value; if not, explain the reason.
13. Let point $A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right)$. Then
$$
x_{1}+x_{2}=\frac{2 k}{1-k^{2}} \text {. }
$$

Let the midpoint of line segment $A B$ be $M$. Then
$$
\begin{array}{l}
x_{M}=\frac{x_{1}+x_{2}}{2}=\frac{k}{1-k^{2}}, \\
y_{M}=k x_{M}+1=\frac{1}{1-k^{2}} .
\end{array}
$$

Let the line $l$ intersect the $y$-axis at point $Q(0, b)$.
Since $P(-2,0), Q(0, b), M\left(\frac{k}{1-k^{2}}, \frac{1}{1-k^{2}}\right)$ are collinear, we have
$$
\begin{array}{l}
\frac{b-0}{0-(-2)}=\frac{\frac{1}{1-k^{2}}-0}{\frac{k}{1-k^{2}}-(-2)} \\
\Rightarrow b=\frac{1}{-k^{2}+\frac{k}{2}+1} .
\end{array}
$$

Since $x_{1}$ and $x_{2}$ are the two distinct negative real roots of the quadratic equation in $x$
$$
\left(1-k^{2}\right) x^{2}-2 k x-2=0
$$

we have
$$
\left\{\begin{array}{l}
x_{1}+x_{2}=\frac{2 k}{1-k^{2}}0, \\
\Delta=4 k^{2}+8\left(1-k^{2}\right)>0 .
\end{array}\right.
$$

Solving this, we get $1<k<\sqrt{2}$.
Thus, the range of $b$ is
$$
(-\infty,-(2+\sqrt{2})) \cup(2,+\infty) \text {. }
$$

Therefore, the maximum length of line segment $C D$ exists, and
$$
|C D|_{\max }=4+\sqrt{2} \text {. }
$$