
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 \[ 10 + 10r + 10r^2 + 10r^3 + \cdots . \] Find \( T(a) + T(-a) \) if \( a \) between \( -1 \) and \( 1 \) satisfies \( T(a)T(-a) = 1200 \).

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