
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
Let the function \( f(x)=\ln x+a\left(\frac{1}{x}-1\right), a \in \mathbf{R} \), and the minimum value of \( f(x) \) is 0. 
1. Find the value of \( a \).
2. Given the sequence \( \left\{a_{n}\right\} \) such that \( a_{1}=1, a_{n+1}=f\left(a_{n}\right)+2\) (where \( n \in \mathbf{N}_{+} \)), and let \( S_{n}=\left[a_{1}\right]+\left[a_{2}\right]+\cdots+\left[a_{n}\right] \), where \( [m] \) denotes the greatest integer less than or equal to the real number \( m \), find \( S_{n} \).

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