Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that the following conditions hold:

1. $f(1) = 1.$

2. $\frac{(x + y)}{2} < f(x + y) \le f(x) + f(y)$ for all $x, y \in \mathbb{N}$.

3. $f(4n + 1) < 2f(2n + 1)$ for all $n \ge 0.$

4. $f(4n + 3) \le 2f(2n + 1)$ for all $n \ge 0.$

Find the sum of all possible values of $f(2023)$.

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