
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 $f(x)=\ln x+\frac{1-x}{ax}$ ($a\in \mathbb{R}$ and $a\neq 0$) and $g(x)=(b-1)x-xe^{x}-\frac{1}{x}$, where $b\in \mathbb{R}$. Determine:

$(Ⅰ)$ Discuss the extreme values of the function $f(x)$.

$(Ⅱ)$ When $a=1$, if $f(x)+g(x)\leqslant -2$ holds for all $x\in (0,+\infty)$, find the range of real numbers for $b$.

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