
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 $f(x)=x-1-a\ln x$ and $g(x)=\frac{x}{{e}^{x-1}}$.  
$(1)$ Discuss the monotonicity of the function $f(x)$.  
$(2)$ If $a \lt 0$ and for any $x_{1}$, $x_{2}\in \left[3,4\right]$ (where $x_{1} \lt x_{2}$) it holds that $\frac{f(x_1)-f(x_2)}{g(x_1)-g(x_2)} \gt -\frac{1}{g(x_1)g(x_2)}$, find the minimum value of the real number $a$.

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