A Recursive Sum

Sequences $\{a_n\}_{n \in \mathbb{W}}$ and $\{b_n\}_{n \in \mathbb{W}}$ are defined as follows:

$a_0 = \frac{1}{2}$ $b_0 = \frac{1}{3}$ $a_{n} = a_0a_{n-1} - b_0b_{n-1} \hspace{5 mm} (\forall n \in \mathbb{N})$ $b_{n} = a_0b_{n-1} + b_0a_{n-1} \hspace{5 mm} (\forall n \in \mathbb{N})$

$A = \sum_{n=0}^{\infty} a_n$ $B = \sum_{n=0}^{\infty} b_n$

If $\frac{A}{B}$ can be expressed as $\frac{p}{q}$ where $p$ and $q$ are coprime natural numbers, enter the value of $p+q$ .