JBMO-SL 2020 A1

Find all triples $(a, b, c)$ of real numbers such that the following system holds:

$$\left\{\begin{array}{l} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\ a^{2}+b^{2}+c^{2}=\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}} \end{array}\right.$$