
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 \( x \in \mathbf{R}, n \in \mathbf{N}^{*} \), define \( C_{x}^{n}=\frac{x(x-1) \cdots(x-n+1)}{n!} \). Let \( P(x) \) be a 6th-degree polynomial that satisfies \( P(0)=1 \) and \( P(k)=2^{k-1} \) for \( k=1,2, \cdots, 6 \). Express \( P(x) \) using \( C_{x}^{k} \) for \( k=1,2, \cdots, 6 \).

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