
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
Find the largest real number \( k \) such that there exists a sequence of positive real numbers \(\{a_i\}\) for which \(\sum_{n=1}^{\infty} a_n\) converges but \(\sum_{n=1}^{\infty} \frac{\sqrt{a_n}}{n^k}\) does not.

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