526 ** The 10 complex roots of the equation $x^{10}+(13 x-1)^{10}=0$ are $r_{1}, \overline{r_{1}}, r_{2}, \overline{r_{2}}$, $r_{3}, \overline{r_{3}}, r_{4}, \overline{r_{4}}, r_{5}, \overline{r_{5}}$. Find the value of the expression
$$
\frac{1}{r_{1} \overline{r_{1}}}+\frac{1}{r_{2} \overline{r_{2}}}+\cdots+\frac{1}{r_{5} \overline{r_{5}}}
$$
Let $\varepsilon=\cos \frac{\pi}{10}+\mathrm{i} \sin \frac{\pi}{10}$, then $\varepsilon^{10}=-1$. From the equation $(13 x-1)^{10}=-x^{10}$, we can set $13 r_{k}-1-r_{k} \cdot \epsilon^{2 k-1}, k=1,2, \cdots, 5$. Thus, $\frac{1}{r_{k}}=13-\varepsilon^{2 k-1}$. Therefore, we have
$$
\begin{aligned}
\sum_{k=1}^{5} \frac{1}{r_{k} \overline{r_{k}}} & =\sum_{k=1}^{5}\left(13-\varepsilon^{2 k-1}\right)\left(13-\bar{\varepsilon}^{2 k-1}\right)=\sum_{k=1}^{5}\left[170-13\left(\varepsilon^{2 k-1}+\bar{\varepsilon}^{2 k-1}\right)\right] \\
& =850-13 \sum_{k=1}^{5}\left(\varepsilon^{2 k-1}+\bar{\varepsilon}^{2 k-1}\right) \\
& =850-26\left(\cos \frac{\pi}{10}+\cos \frac{3 \pi}{10}+\cos \frac{5 \pi}{10}+\cos \frac{7 \pi}{10}+\cos \frac{9 \pi}{10}\right)=850 .
\end{aligned}
$$

Thus, the value of the required algebraic expression is 850.