
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 \( f(x) = \frac{x}{x^{2} + 3x + 2} \). The point \( A_{0} \) represents the origin, and the coordinates of point \( A_{n} \) are \( (n, f(n)) \) for \( n \in \mathbb{N}^{*} \). Let \( k_{n} \) represent the slope of the line \( A_{0}A_{n} \). Define \( S_{n} = k_{1} + k_{2} + \cdots + k_{n} \). Find \( S_{n} \).

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