
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 function $f(x) = \ln x$.  
$(1)$ Discuss the monotonicity of $g(x) = f(x) - ax$;  
$(2)$ Let $F(x) = (x-2)e^{x} + f(x) - x - m$, $m \in \mathbb{Z}$. If $F(x) \leq 0$ holds for all $x \in \left(\frac{1}{4}, 1\right)$, find the minimum value of $m$.

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