GAIC Math 2024 Problem 25

For a natural number $n$, let $K(n,0)=\varnothing$. For any non-negative integers $m$ and $n$, define $K(n,m+1)$ as the set of elements $k$ such that $1\leq k\leq n$ and $K(k,m)\cap K(n-k,m)=\varnothing$, then the set $K(2004,2004)$ contains \underline{\hspace{2cm}} elements.