
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
Let $X$ be a discrete random variable with a uniform distribution on $\{0,1, \ldots, n\}$, that is, $\mathrm{P}(X=k)=\frac{1}{n+1}$ where $k=0,1, \ldots, n$. By showing that $G(s)=\mathrm{E} s^{X}=\frac{1}{n+1} \frac{1-s^{n+1}}{1-s}$, and finding $\mathrm{E} X$ and $\mathrm{E} X^{2}$, establish the well-known formulas

$$
\sum_{k=1}^{n} k=\frac{n(n+1)}{2}, \quad \sum_{k=1}^{n} k^{2}=\frac{n(n+1)(n+2)}{6}
$$

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