Compute the sum of the series \( \sum_{k=0}^{\infty} \frac{1}{(4 k+1)(4 k+2)(4 k+3)(4 k+4)}=\frac{1}{1 \cdot 2 \cdot 3 \cdot 4}+\frac{1}{5 \cdot 6 \cdot 7 \cdot 8}+\cdots  \). The original answer is in a special format：$\frac{\ln m}n-\frac\pi{p}.$ provide the sum of the denominators in the expression. 

Let's think step by step and output the final answer within \boxed{}.
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