
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 \(-1 < r < 1\), let \( T(r) \) denote the sum of the geometric series 
\[ 20 + 20r + 20r^2 + 20r^3 + \cdots. \]
Let \( b \) between \(-1\) and \(1\) satisfy \( T(b)T(-b) = 4000 \). Find \( T(b) + T(-b) \).

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