Let $m = 2007^{2008}$. How many natural numbers $n$ are there such that $n < m$ and $n(2n + 1)(5n + 2)$ is divisible by $m$ (i.e., $m \mid n(2n + 1)(5n + 2)$)?

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