
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 where $u_n = n^4 + n^2 + n$, define $\Delta^1(u_n) = u_{n+1} - u_n$ and for any integer $k > 1$, let $\Delta^k(u_n) = \Delta^1(\Delta^{k-1}(u_n))$. Determine the smallest $k$ such that $\Delta^k(u_n) = 0$ for all $n$.
**A)** $k=4$
**B)** $k=5$
**C)** $k=6$
**D)** $k=7$
**E)** for no value of $k$

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