
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
Given the function \( f(x) = a_{1} x + a_{2} x^{2} + \cdots + a_{n} x^{n} \) (\( n \in \mathbf{N}_{+} \)), with \( a_{1}, a_{2}, \cdots, a_{n} \) as a sequence, and knowing that \( f(1) = n^{2} + 1 \):

1. Find the general term formula for the sequence \( \left\{a_{n}\right\} \).
2. Find \( \lim _{n \rightarrow \infty}\left(1-\frac{1}{a_{n}}\right)^{n} \).

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