$\large{S = 1 + \dfrac{1}{5}- \dfrac{1}{7} - \dfrac{1}{11} + \dfrac{1}{13} + \dfrac{1}{17} - \dfrac{1}{19} - \dfrac{1}{23} + \ldots }$

If $S$ can be expressed as $\dfrac{\pi^A}{B}$ for positive integers $A,B$, submit the value of $A+B$ as your answer.