
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 \( u_{n} = n^{3} + n \), and the definitions of \(\Delta^{\prime}(u_{n}) = u_{n+1} - u_{n}\) and \(\Delta^{k}(u_{n}) = \Delta^{\prime}[\Delta^{k-1}(u_{n})]\) for integers \( k > 1 \), determine for which value of \( k \) the condition \(\Delta^{k}(u_{n}) = 0\) holds for all \( n \).

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