
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 $f(x)=\ln x-x$, $h(x)=\frac{a{e}^{x}}{x}+1$, where $a\in R$.  
$(1)$ Find the equation of the tangent line passing through the origin for the curve $y=f(x)$.  
$(2)$ If $f(x)+h(x)\geqslant 0$ holds for all $x\in [1,+\infty)$, find the range of values for $a$.

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