
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 $T(r)$ represent the sum of the geometric series \[20 + 20r + 20r^2 + 20r^3 + \cdots.\] Determine $a$ between $-1$ and $1$ such that $T(a)^2 \cdot T(-a)^2 = 640000$. Find $T(a) + T(-a)$.

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