
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 $$f(x)=x+a\ln(x) \quad (a\in \mathbb{R}), \quad g(x)=\frac{e^{x-1}}{x}-1.$$  
(I) If the line $y=0$ is tangent to the graph of $y=f(x)$, find the value of $a$.  
(II) Let $a>0$, and for any $x_1$, $x_2 \in [3,+\infty)$ ($x_1\neq x_2$), it holds that $|f(x_1)-f(x_2)|<|g(x_1)-g(x_2)|$. Determine the range of the real number $a$.

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