
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 \( a_{n} \) is defined as follows: \( a_{1}=0, a_{2}=1 \)
$$
a_{n}=\frac{1}{2} n a_{n-1}+\frac{1}{2} n(n-1) a_{n-2}+(-1)^{n}\left(1-\frac{n}{2}\right) \quad \text{for} \quad n \geq 3
$$

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

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