That seems nice! (Updated)

$\Large{ A = 1! \cdot 2! \cdot 3! \cdots 16! }$

Find the number of all positive cubic divisors of $A$ . If the answer is $p$ , enter the remainder of $p$ when divided by $1000$ .

Assumption : A cubic divisor of $A$ is a divisor of $A$ that is also a cubic number; that is, a perfect third power.