200 Followers Problem (repost)

Calculus Level 4

\[ \large \displaystyle \sum_{n=0}^{\infty} \dfrac{(-1)^n}{2n+1}\cdot \displaystyle \sum_{n=0}^{\infty} \left [ \dfrac{(-1)^n}{4n+1}+\dfrac{(-1)^n}{4n+3} \right] \] If the value of above expression can be expressed in the form \(\dfrac{\pi^{a}}{b\sqrt{c}}\), find the value of \(a+b+c\).

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