Find the sum

Calculus Level pending

\[\begin{align} S&=\frac{1}{1\cdot2}+\frac{2}{2\cdot3}+\frac{3}{3\cdot4}+\frac{4}{4\cdot5}+\frac{0}{5\cdot6}+\frac{1}{6\cdot7}+...\\ &=\sum_{k=1}^∞\frac{k\text { mod 5}}{k(k+1)}\end{align}\]

Evaluate the infinite sum above and enter \(\lfloor 10^{10} S \rfloor\).

Notation: \(\lfloor \cdot \rfloor\) denotes the floor function.

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