Two linear functions $f(x)$ and $g(x)$ satisfy the following properties for all $x$:

- $f(x) + g(x) = 2$
- $f(f(x)) = g(g(x))$

Additionally, $f(0) = 2022$. Compute $f(1)$.

Let's think step by step and output the final answer within \boxed{}.
user