
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
The sequence $\left\{a_{n}\right\}$ is defined as follows: $a_{1} = 0$, $a_{2} = 1$, and for $n \geq 3$, \( a_{n} = \frac{1}{2} n a_{n-1} + \frac{1}{2} n(n-1) a_{n-2} + (-1)^{n} \left(1-\frac{n}{2}\right) \). Find the simplest expression for \( f_{n} = \sum_{k=0}^{n-1} (k+1) C_{n}^{k} a_{n-k} \).

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