Question 80
Let $N_0$ be the set of nonnegative integers and $\mathbb{Z}$ be the set of integers. Find the number of possible tuples $(f(0), f(1), \dots, f(2024))$ for $f: N_0 \to \mathbb{Z}$ satisfying
$$
f(m+n)^2 = f(m|f(n)|) + f(n^2)
$$
for any $m, n \in N_0$.
$$
f(m+n)^2 = f(m|f(n)|) + f(n^2)
$$
for any $m, n \in N_0$.