Find positive numbers $n$ and $a_1, \dots, a_n$ such that $a_1 + \dots + a_n = 1000$ and $a_1a_2 \cdots a_n$ is as large as possible.

Question

Find positive numbers $n$ and $a_1, \dots, a_n$ such that $a_1 + \dots + a_n = 1000$ and $a_1a_2 \cdots a_n$ is as large as possible.

This is a problem from the book "problem-solving through problems" and the key heuristic tried to communicate here is that the choice of $1000$ can be replaced by $2,3,4,5,6,7,8,9, \dots$ to help finding some patterns in the problem.

They conclude that

1. no $a_i$ will be greater than $4$,
2. no $a_i$ will be equal $1$,
3. all $a_i$'s can be taken to be $2$ or $3$,
4. at most two $a_i$'s will be equal $2$.

Then somehow from these conclusions they end up with the maximum being $3^{332} \cdot 2^2$.

I cannot conclude any of these points they make. If $n = 2$ and $a_1+a_2 = 9$, then surely $a_1 = 4$ and $a_2 =5$ is a solution and already the first point fails?

While this seems like a good book the I think that heuristic here is not being communicated very well. Is there some general pattern one can derive from considering the cases with $2,3,4,5,6,7,8,9, \dots$?