Let $\alpha = \cos^{-1} \left( \frac{3}{5} \right)$ and $\beta = \sin^{-1} \left( \frac{3}{5} \right)$. 

\[\sum_{n=0}^{\infty}\sum_{m=0}^{\infty} \frac{\cos(\alpha n +\beta  m)}{2^n3^m}\]

can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A +B$.

Let's think step by step and output the final answer within \boxed{}.
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