
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
Let \( T(r) \) denote the sum of the geometric series \( 15 + 15r + 15r^2 + 15r^3 + \cdots \) for \( -1 < r < 1 \). Let \( b \) between \( -1 \) and \( 1 \) satisfy \( T(b)T(-b) = 2700 \). Find \( T(b) + T(-b) \).

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