
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 functions $f(x) = 5 + \ln x$ and $g(x) = \frac{kx}{x + 1}$, where $k \in \mathbb{R}$:

1. Find the value of $k$ such that the tangent line to the function $f(x)$ at the point $(1, f(1))$ is tangent to the graph of the function $y = g(x)$.
2. If $k \in \mathbb{N}^{*}$ and $x \in (1, +\infty)$, find the maximum value of $kl$ such that $f(x) > g(x)$ always holds. (Note: $\ln 5 \approx 1.61$, $\ln 6 \approx 1.7918$, and $\ln (\sqrt{2} + 1) = 0.8814$.)

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