
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
Define
\[ A = \frac{1}{1^3} + \frac{1}{5^3} - \frac{1}{7^3} - \frac{1}{11^3} + \frac{1}{13^3} + \frac{1}{17^3} - \dotsb, \] 
which omits all terms of the form \(\frac{1}{n^3}\) where \(n\) is an odd multiple of 3, and 
\[ B = \frac{1}{3^3} - \frac{1}{9^3} + \frac{1}{15^3} - \frac{1}{21^3} + \frac{1}{27^3} - \frac{1}{33^3} + \dotsb, \]
which includes only terms of the form \(\frac{1}{n^3}\) where \(n\) is an odd multiple of 3.

Determine \(\frac{A}{B}\).

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