
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 $f(x)=ax+\ln x$, where $a$ is a constant, and $e$ is the base of the natural logarithm.
$(1)$ When $a=-1$, find the maximum value of $f(x)$;
$(2)$ Let $g(x)=xf(x)$, $h(x)=2ax^{2}-(2a-1)x+a-1$. If for $x\geqslant 1$, $g(x)\leqslant h(x)$ always holds, find the range of the real number $a$.

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