
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 
\[ A = \frac{1}{1^2} + \frac{1}{3^2} - \frac{1}{5^2} - \frac{1}{7^2} + \frac{1}{9^2} + \frac{1}{11^2} - \dotsb, \]
which omits all terms of the form \(\frac{1}{n^2}\) where \(n\) is a multiple of 4, and 
\[ B = \frac{1}{4^2} - \frac{1}{8^2} + \frac{1}{12^2} - \frac{1}{16^2} + \frac{1}{20^2} - \frac{1}{24^2} + \dotsb, \]
which includes only terms of the form \(\frac{1}{n^2}\) where \(n\) is a multiple of 4.

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

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