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$\large \dfrac{1}{1}+\dfrac{1}{2}-\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}-\dfrac{1}{7}-\dfrac{1}{8}+\cdots$

If the closed form of the sum above can be expressed as $\dfrac{\pi+\ln (a)}{a}$, where $a$ is a positive integer, find $a$.

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