
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
(1) Simplify the expression $\sum_{k=0}^{n} (-1)^k C_{n}^{k} \cdot 2^{n-k}$ for $n\in\mathbb{N}^*$;

(2) Simplify the expression $\sum_{k=0}^{n} (C_{n}^{k})^2$ using the identity $(1+x)^{2n} = (1+x)^n \cdot (1+x)^n$.

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