
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
Assuming an annual interest rate of \( i \), calculated with compound interest, how much principal is needed to be able to withdraw 1 dollar at the end of the first year, 4 dollars at the end of the second year, \(\cdots\), \( n^2 \) dollars at the end of the \( n \)-th year, and continue doing so forever? (From the 18th Putnam competition)

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