
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 $f(x) = k(x-1) - 2\ln x$ ($k > 0$).

(1) If the function $f(x)$ has exactly one zero point, find the value of the real number $k$;

(2) Let the function $g(x) = xe^{1-x}$ (where $e$ is the base of the natural logarithm), if for any given $s \in (0, e)$, there exist two different $t_i \in \left( \frac{1}{e^2}, e \right)$ ($i=1,2$), such that $f(t_i) = g(s)$ holds, find the range of values for the real number $k$.

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