
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

Find the following differences:

a) $\Delta n^{2}$

b) $\Delta n(n-1)$

c) $\Delta n^{\mathrm{k}}$

d) $\Delta C_{\mathrm{n}}^{\mathrm{k}}$

Find the sequence $\left\{a_{n}\right\}$ such that $\Delta a_{n}=n^{2}$. Using the result from the previous problem, derive the formula for the sum $1^{2}+2^{2}+3^{2}+\ldots+n^{2}$.

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