
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
For a function $F(x)$ defined on $D$, if there exists $x_{0} \in D$ such that $F(x_{0}) = x_{0}$, then $x_{0}$ is called a fixed point of $F(x)$. Let $f(x) = (x-1)e^{x} - a\ln x + x$. Given that $x_{0} (x_{0} \neq 1)$ is a fixed point of the function $f(x)$, find:<br/>
$(1)$ The range of real numbers for $a$;<br/>
$(2)$ If $k \in \mathbb{Z}$ and $kx_{0} < a$ holds for any $x_{0}$ satisfying the condition, find the maximum value of the integer $k$.<br/>
(Reference data: $\ln 2 \approx 0.693$, $\ln 3 \approx 1.1$, $e^{\frac{2}{3}} \approx 1.95$, $e^{2} \approx 7.39$, $e^{\frac{3}{2}} \approx 4.48$)

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