
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 \(v_n = n^4 + 2n^2 + 2\). Define the sequence of differences \(\Delta^k(v_n) = \Delta^1(\Delta^{k-1}(v_n))\). Identify for which \(k\) the sequence \(\Delta^k(v_n)\) becomes zero for all \(n\).

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