
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
$$
f(x)=\ln x+a\left(\frac{1}{x}-1\right)(a \in \mathbf{R}),
$$

and the minimum value of $f(x)$ is 0.

(1) Find the value of $a$.

(2) If the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=1, \quad a_{n+1}=f\left(a_{n}\right)+2 \quad (n \in \mathbf{Z}_{+}),
$$

let $S_{n}=\left[a_{1}\right]+\left[a_{2}\right]+\cdots+\left[a_{n}\right]$, where $[m]$ denotes the greatest integer not exceeding the real number $m$. Find $S_{n}$.

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