
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
Suppose $x>1$, and let $f(x) = \ln x - \frac{1}{2}(x-1)^{2}$.  
Determine the interval on which $f(x)$ is increasing.  
Find all possible values of $k$ such that for $x>1$, $f(x) > k(x-1)$ holds for all $x$ in the interval $(1, x_0)$.

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