15. Find the best constant $k$ that satisfies the following inequality:
$$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}+k \cdot \frac{y z+z x+x y}{x^{2}+y^{2}+z^{2}} \geqslant 3+k \text {, }$$

where $x, y, z, k$ are positive real numbers.
15. The best constant $k=3 \sqrt[3]{4}-2 \approx 2.7622$, equality holds when:
$$\begin{aligned}
x & =\frac{1}{3}+\sqrt[3]{2}-\frac{\sqrt[3]{4}}{3}+\frac{2}{3} \sqrt{\sqrt[3]{4}+8 \sqrt[3]{2}-11} \cos \left(\frac{1}{3} \arccos \sqrt{\frac{17-3 \sqrt[3]{4}}{20}}\right) \\
& \approx 1.5949 \\
y & =\frac{2}{3} \sqrt[3]{4}+\frac{2}{3} \sqrt{3 \sqrt[3]{4}-\sqrt[3]{2}-3} \sin \left(\frac{1}{3} \arccos \sqrt{\frac{27+27 \sqrt[3]{2}-27 \sqrt[3]{4}}{20}}\right) \\
& \approx 1.1067 \\
z & =1
\end{aligned}$$