
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
A sequence \( a_{n} \) is defined as follows: \( a_{1}=0 \), \( a_{2}=1 \)
\[ a_{n}=\frac{1}{2} n a_{n-1}+\frac{1}{2} n(n-1) a_{n-2}+(-1) n\left(1-\frac{n}{2}\right) \text{ for } n \geqslant 3 \]

Find the simplest expression for \( f_{n}=a_{n}+2 C_{n}^{1} a_{n-1}+3 C_{n}^{n} a_{n-2}+\cdots+(n-1) C_{n}^{n-2} a_{2}+n C_{n}^{n-1} a_{1} \).

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