
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let  $\alpha = \cos^{-1} \left( \frac35 \right)$  and  $\beta = \sin^{-1} \left( \frac35 \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  $1000A +B$ .

Present the answer in LaTex format: \boxed{Your answer}
user