
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 \(\left\{a_{n}\right\}\) such that
\[ a_{0}=\frac{1}{k}, a_{n}=a_{n-1}+\frac{1}{n^{2}} a_{n-1}^{2} \quad (k \in \mathbf{Z}_{+}), \]
if \( a_{n} < 1 \) for all \( n \in \mathbf{Z}_{+} \), determine the range of values for \( k \).

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