
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 any positive integers \(m\), \(n\) and any complex number \(z\), we have
$$
\begin{aligned}
& \sum_{i=0}^{n}(-1)^{i} C_{n}^{i}(z+n-i)(z+n-i-1) \cdots(z+n-i-m+1) \\
= & \left\{\begin{array}{l}
0 \quad(m<n), \\
m!\quad(m=n), \\
C_{m}^{n} \cdot n!\cdot z(z-1)(z-2) \cdots(z-m+n+1) \quad(m>n) .
\end{array}\right.
\end{aligned}
$$

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