
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 functions $f(x)=\ln x-(a-1)x+1$ and $g(x)=x(e^{x}-1)$, where $a\in \mathbb{R}$. Find:

$(1)$ The extreme value of $f(x)$.

$(2)$ If $g(x)\geq f(x)$, find the range of real number $a$.

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