How to calculate sums like $\sum_{n=2}^{\infty} \frac{ sinc (-4 \cdot \pi (n-3)(n-4) ) }{n(n-1)} $ with the Residue Theorem?

Question

I'm trying to compute sums like $$S:= \sum_{n=2}^{\infty} \frac{ \operatorname{sinc} (-4\pi (n-3)(n-4) ) }{n(n-1)} $$ by means of the Residue Theorem, which states that $$\lim_{k \to +\infty} \sum_{k=-N}^{N} f(k) = - \{ \text{sum of the residues of } \pi f(z) \cot(\pi z) \text{ at the poles of }f(z) \} .$$ Here, $\operatorname{sinc}(x) = \sin(x)/x $ at $x \in \mathbb{C} \setminus \{0\} $, and $\operatorname{sinc}(x) = 1$ at $x = 0$.

I'm having some trouble determining the nature and degree of the poles, and how to use the cotangent function as well. Considering the definition of the sinc function, we could rewrite the sum as follows:

$$S = \sum_{n=2}^{\infty} \frac{\sin(-4\pi (n-3)(n-4))}{-4 \pi n(n-1)(n-3)(n-4)} ,$$ so if we'd take $$f(z) = \frac{\sin(-4\pi(z-3)(z-4))}{-4 \pi z(z-1)(z-3)(z-4)}, $$ we can use the Residue Theorem to compute $S$, as $f$ amounts to zero when $k \in \mathbb{Z}_{< 0}$. Thus, I would guess that the poles of $f$ are at $z=0, z=1, z=3,$ and $z=4$, and are all of order $1$.

However, I'm not quite sure how to calculate the residues when the cotangent function is involved. I know the residues of $\pi f(z) \cot(\pi z) $ amount to $\frac{7}{12}, -\frac{10}{12}, \frac{2}{12},$ and $\frac{1}{12} $ at $z=0,1,3,4$ respectively by calculating them in WolframAlpha, but I'm not entirely sure how the software computes them. Does it use the Taylor series of the cotangent function for this? How can these residues be calculated by hand?

Moreover, the sum of the residues as described above amounts to zero. However, I know it must be the case that $S=\frac{1}{4}$, as the sum only has nonzero terms when the arguments of the sinc function are zero, which happens at $n=3$ and $n=4$. Thus, $S=\frac{1}{6} + \frac{1}{12} = \frac{1}{4}$.

I do see that the sum of the residues at $z=0$ and $z=1$ give the correct evaluation for $S$, but then I fail to see why the residues of the other two poles at $z=3$ and $z=4$ must be discarded.

Questions:

1. What exactly are the poles of $f$, and what are their degrees?
2. How do I take into account the $\cot(\pi z)$ function when computing the residues?