6. (She Di Province) Let $S$ be the sum of the lengths of two chords on the unit circle that are $d$ (a real number) apart. Try to find the range of values for $S$.

保留源文本的换行和格式，直接输出翻译结果如下：

6. (She Di Province) Let $S$ be the sum of the lengths of two chords on the unit circle that are $d$ (a real number) apart. Try to find the range of values for $S$.
Solve: First, when $d>2$,
it is meaningless;
$$
\begin{array}{l}
\text{When } d=2, S=0, \\
\because \text{When } 10-d<2, \text{ set }
\end{array}
$$

Prevent chord distance from the center $r_{1}, r_{2}$
$$
\begin{array}{l}
\left(\left|r_{1} \pm r_{2}\right|=d\right), \text{ then } \\
S=2 \sqrt{1-r_{1}^{2}}+2 \sqrt{1-r_{2}^{2}} \\
=4 \sqrt{1-r_{1}^{2}+1-r_{2}^{2}} \\
=4 \sqrt{1-\left(r_{1} \pm r_{2}\right)^{2}} \\
=2 \sqrt{4-d^{2}} .
\end{array}
$$

Extra, when one chord shrinks to a point, the sum of the two chords $S$ is minimized, in this case
$$
S=2 \sqrt{1-(d-1)^{2}}=2 \sqrt{2 d-d^{2}} .
$$

That is, $2 \sqrt{2 d-d^{2}}<S \div 2 \sqrt{4-d^{2}}$.