Who's Up To the Challenge?83

Define $I_n=\int_0^\infty \cdots \int_{0}^{\infty} \frac{1}{\left( \sum_{k=1}^n \left( x_k+\frac{1}{x_k}\right) \right)^2} \frac{dx_1}{x_1} \cdots \frac{dx_n}{x_n}$

If the limit $\lim_{n \to \infty} \frac{I_n}{n!}$

is equal to $\dfrac{Ae^{-B\gamma}}{C}$ for positive integers $A,B,C$ and coprime integers $A,C$ ,find $A+B+C$ . If the limit diverges enter $0$ .

Notation : $\gamma$ denotes the Euler Mascheroni constant.

Note : Problem is not original.