In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, it is known that the distances from the diagonal $A C_{1}$ to the edges $D D_{1}$, $A_{1} B_{1}$, and $B C$ are $\frac{1}{m}$, $\frac{1}{n}$, and $\frac{1}{p}$, respectively. The length of the diagonal $A C_{1}$ is $\qquad$.
$120 \sqrt{\frac{2}{m^{2}+p^{2}-n^{2}}+\frac{2}{m^{2}+n^{2}-p^{2}}+\frac{2}{n^{2}+p^{2}-m^{2}}}$. The solution is similar to problem 108.
$$
\frac{1}{a^{2}}+\frac{1}{b^{2}}=m^{2} \quad \frac{1}{b^{2}}+\frac{1}{c^{2}}=n^{2} \quad \frac{1}{c^{2}}+\frac{1}{a^{2}}=p^{2}
$$

Therefore,
$$
\begin{array}{l}
\frac{1}{a^{2}}=\frac{1}{2}\left(m^{2}+p^{2}-n^{2}\right), \\
\frac{1}{b^{2}}=\frac{1}{2}\left(m^{2}-p^{2}+n^{2}\right) \\
\frac{1}{c^{2}}=\frac{1}{2}\left(-m^{2}+p^{2}+n^{2}\right) .
\end{array}
$$

The length of the diagonal
$$
\sqrt{a^{2}+b^{2}+c^{2}}=\sqrt{\frac{2}{m^{2}+p^{2}-n^{2}}+\frac{2}{m^{2}-p^{2}+n^{2}}+\frac{2}{-m^{2}+p^{2}+n^{2}}} .
$$