
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 \( f(x) = \ln x + a\left(\frac{1}{x} - 1\right) \), where \( a \in \mathbb{R} \) and the minimum value of \( f(x) \) is 0, 

(1) Find the value of \( a \).
(2) Given the sequence \(\{a_n\}\) satisfying \( a_1 = 1 \) and \( a_{n+1} = f(a_n) + 2 \) for \( n \in \mathbb{N}_+ \), and let \( S_n = \left[a_1\right] + \left[a_2\right] + \left[a_3\right] + \cdots + \left[a_n\right] \), where \([m]\) denotes the greatest integer less than or equal to \( m \). Find \( S_n \).

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