
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 $f(x)=\ln x+(a+1)x+1$, $a\in \mathbb{R}$, and $g(x)=xe^{x}$.  
$(1)$ If the slope of the tangent line to the curve $f(x)$ at the point $(1,f(1))$ is $3$, find the value of $a$.  
$(2)$ For $x\geqslant -2$, if the function $y=g(x)-m+2$ has two distinct zeros, find the range of values for $m$.  
$(3)$ If the inequality $g'(x)-f(x)\geqslant e^{x}$ holds for all $x\in (0,+\infty)$, find the range of real numbers for $a$.

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