
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
Given the sequence \(\left\{a_{n}\right\}\) which satisfies \(a_{0}=\frac{1}{k}\) and \(a_{n}=a_{n-1}+\frac{1}{n^{2}} a_{n-1}^{2}\), where \(k\) is a positive integer. If for all \(n \in \mathbf{N}^{*}\), we always have \(a_{n} < 1\), find the range of values for \(k\).

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