Cute problem with a+b+c = 2022

 $a,b,c$ are real numbers such that

$$a + b + c = 2022$$

and

$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{2022}$$

What are the possible values of 

$$\frac{1}{a^{2023}} + \frac{1}{b^{2023}} + \frac{1}{c^{2023}}$$

Try to keep the algebraic manipulations to a minimum.

[Solution]