
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 $f(x)= \frac {1+\ln x}{x-1}$ and $g(x)= \frac {k}{x}$ ($k\in\mathbb{N}^*$), for any $c > 1$, find the maximum value of $k$ such that there exist real numbers $a$ and $b$ such that $0 < a < b < c$ and $f(c)=f(a)=g(b)$.

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