
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 $a \gt 0$, the function $f\left(x\right)=xe^{x}-ax$.  
$(1)$ When $a=1$, find the equation of the tangent line to the curve $y=f\left(x\right)$ at $x=1$.  
$(2)$ If $f\left(x\right)\geqslant \ln x-x+1$ always holds, find the range of real numbers for $a$.

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