
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
\[A' = \frac{1}{1^2} + \frac{1}{7^2} - \frac{1}{11^2} - \frac{1}{17^2} + \frac{1}{19^2} + \frac{1}{23^2} - \dotsb,\]which omits all terms of the form $\frac{1}{n^2}$ where $n$ is an odd multiple of 5, and
\[B' = \frac{1}{5^2} - \frac{1}{15^2} + \frac{1}{25^2} - \frac{1}{35^2} + \frac{1}{45^2} - \frac{1}{55^2} + \dotsb,\]which includes only terms of the form $\frac{1}{n^2}$ where $n$ is an odd multiple of 5.

Determine $\frac{A'}{B'}.$

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