
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
For $|r| < 1$, consider the sum of the geometric series
\[10 + 10r + 10r^2 + 10r^3 + \cdots.\]
Let $b$ such that $-1 < b < 1$ and $S(b)^2 - S(-b)^2 = 7840$. Find $S(b) - S(-b)$.

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