Question 111
Let $A$ be a set of functions defined for integers from $1$ to $2023$ and taking integer values from $1$ to $2023$. Suppose that $A$ satisfies the following two conditions:
* For any function $f$ belonging to $A$ and any integers $x$ and $y$ with $1 \le x \lt y \le 2023$, we have $f(x) \ge f(y)$.
* For any functions $f$ and $g$ belonging to $A$ and any integer $x$ from $1$ to $2023$, we have $f(g(x)) = g(f(g(x)))$.
In this case, find the maximum possible number of elements of $A$.
* For any function $f$ belonging to $A$ and any integers $x$ and $y$ with $1 \le x \lt y \le 2023$, we have $f(x) \ge f(y)$.
* For any functions $f$ and $g$ belonging to $A$ and any integer $x$ from $1$ to $2023$, we have $f(g(x)) = g(f(g(x)))$.
In this case, find the maximum possible number of elements of $A$.