
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
Show the following identities for all integers $n > 0$:

$$
\begin{aligned}
\sum_{k=0}^{n}\binom{n}{k}=2^{n} & \sum_{k=0, k \text { even }}^{n}\binom{n}{k}=2^{n-1} \\
\sum_{k=0}^{n} k\binom{n}{k}=n 2^{n-1} & \sum_{k=0}^{n} \sum_{\ell=0}^{k}\binom{n}{k}\binom{k}{\ell}=3^{n}
\end{aligned}
$$

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