$\large{S = \left( 1 + \dfrac{1}{2} + \dfrac{1}{3} \right) - \left( \dfrac{1}{4} + \dfrac{1}{5} + \dfrac{1}{6} \right) + \left( \dfrac{1}{7} + \dfrac{1}{8} + \dfrac{1}{9} \right) - \left( \dfrac{1}{10} + \dfrac{1}{11} + \dfrac{1}{12} \right) + \ldots }$

If $S$ can be expressed as:

$\large{\dfrac{A\sqrt{B}}{C}\pi^D + \dfrac{\ln(E)}{F} }$

for positive integers $A,B,C,D,E,F$ where $\gcd(A,C)=1$ and $B,E$ aren't any $m^{th}$ power of a positive integer with $m \in \mathbb Z, \ m \geq 2$.

Submit the value of $A+B+C+D+E+F$ as your answer.