
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
Consider a sequence defined by \(u_n = n^3 - n\). Further, define \(\Delta^1(u_n)=u_{n+1}-u_n\) and for any integer \(k>1\), define \(\Delta^k(u_n)=\Delta^1(\Delta^{k-1}(u_n))\). Determine for what smallest value of \(k\), \(\Delta^k(u_n)=0\) for all \(n\).
**A) \(k=1\)**  
**B) \(k=2\), but not if \(k=1\)**  
**C) \(k=3\), but not if \(k=2\)**  
**D) \(k=4\), but not if \(k=3\)**  
**E) for no value of \(k\)**

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