
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 $f(x)=\frac{1}{2}{e}^{2x}+(a-2){e}^{x}+(1-a)x(a∈R)$.  
$(1)$ When $a=-1$, find the equation of the tangent line to the graph of $F\left(x\right)=f\left(x\right)+x$ at the point $\left(0,F\left(0\right)\right)$;  
$(2)$ If the graph of $f\left(x\right)$ intersects the line $y=1$ at exactly two distinct points, find the range of real numbers for $a$.

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