Saddle Point Approximation for Multiple Contour Integrals

Using the multinomial theorem
$$ left(sum_{j=1}^{N} x_j right)^N = sum_{m_1 + cdots + m_n = N} frac{N!}{m_1! cdots m_N!} x^{m_1} cdots x^{m_N}, $$
and the contour integral expression of the Kronecker delta
$$ delta(k, ell) = frac{1}{2pi i}oint frac{dz}{z} z^{k-ell}, $$
one can show
$$ N! = frac{1}{(2pi i)^N} oint left[prod_{ell=1}^N frac{dq_{ell}}{q_{ell}^2}right] left(sum_{j=1}^{N} q_{j}right)^{N},$$
where we are applying $N$ contour integrations in sequence.

Question: How can we apply a saddle point approximation to the multiple contour integrals to derive Stirling’s approximation (i.e., $N!simeq sqrt{2pi N} (N/e)^N$)?