
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:
$$
\begin{array}{r}
a_{1}=0, a_{2}=1, \cdots, a_{n}=\frac{1}{2} n a_{n-1}+\frac{1}{2} n(n-1) a_{n-2}+(-1)^{n}\left(1-\frac{n}{2}\right) \\
(n \geqslant 3).
\end{array}
$$

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

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