
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
Let  $k$  be an integer greater than  $1.$  Suppose  $a_{0}>0$  and define
\[a_{n+1}=a_{n}+\frac1{\sqrt[k]{a_{n}}}\]
for  $n\ge 0.$  Evaluate
\[\lim_{n\to\infty}\frac{a_{n}^{k+1}}{n^{k}}.\]

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