
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: $a_{1}=0, a_{2}=1, a_{n}=\frac{1}{2} n a_{n-1} + \frac{1}{2} n(n-1) a_{n-2} + (-1)^{n}\left(1-\frac{n}{2}\right)$ for $n \geqslant 3$. Find the simplest expression for $f_n = a_n + 2 \binom{n}{1} a_{n-1} + 3 \binom{n}{2} a_{n-2} + \cdots + (n-1) \binom{n}{n-2} a_2 + n \binom{n}{n-1} a_1$.

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