
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 $f(x) = \frac{e^x}{e^x - 1}$, where \( x > 0 \):
1. Find the equation of the tangent line to the graph of \( y = f(x) \) at the point \( (\ln 2, f(\ln 2)) \).
2. Let \( g(x) = \frac{k}{x + 1} \), where \( x > 0 \) and \( k \in \mathbb{N}^* \). If \( f(x) > g(x) \) holds true for all values within their domain, find the maximum value of \( k \).

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