
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 $f(x)=2x-a$ and $g(x)=\ln x$.  
$(1)$ If point $M$ is a point on the graph of $y=f(x)$ and point $N$ is a point on the graph of $y=g(x)$, find the minimum distance between points $M$ and $N$ when $a=-1$.  
$(2)$ If the function $h(x)=-f(x) \cdot g(x)$ is monotonically decreasing on $(0, +\infty)$, find the range of real numbers for $a$.

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