
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
For \( x \in \mathbb{R} \) and \( n \in \mathbb{N}^{*} \), define
\[ \mathrm{C}_{x}^{n} = \frac{x(x-1) \cdots (x-n+1)}{n!}. \]
Let \( p(x) \) be a polynomial of degree \( 2m \) such that \( p(0) = 1 \) and \( p(k) = 2^{k-1} \) for \( k = 1, 2, \ldots, 2m \). Express \( p(x) \) using \( \mathrm{C}_{x}^{k} \) for \( k = 1, 2, \ldots, 2m \).

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