AM-GM Doesn't work (or does it ?)

let $a_1,a_2,a_3,...,a_n$ is a set of positive integer ( not sequence) that do not have to different each other so : $a_1+a_2+a_3+...+a_n = 1000$ what is the maximum value of: $a_1 * a_2 * a_3 * ... * a_n ?$

note:

$a_1,a_2,...,a_n$ can be just [1000] , or can be [1,999] ,or [250,250,250,250] , or even [1,1,2,3,1,2,1,2,3,1,2,1,2,3,2,3,1,.....,1] so,the number of $a_1,a_2,...,a_n$ is not fixed.

#NumberTheory

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$