
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
The geometric series \(a + ar + ar^2 + \cdots\) has a total sum of \(15\), and the terms involving square powers of \(r\) (\(ar^0, ar^4, ar^9,\ldots\)) have a sum of \(9\). What is \(r\)?

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