\[ \Large {\prod_{r=1}^{23}} \large{ \text{exp} \left [ \displaystyle \int_0^\infty \dfrac{e^{-\left(\frac{r-24}{24} \right)t} - \left(\frac{r-24}{24} \right)e^{-t} -1 + \left(\frac{r-24}{24} \right)}{t (e^t-1)} \, dt \right ]} \]

If the product above equals to \(\displaystyle \sqrt{ \dfrac{(A\pi)^B}{C} } \) for positive integers \(A,B\) and \(C\), find the value of \(A+B+C\).

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