
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 $f\left(x\right)=e^{x}-ax$.  
$(1)$ If $a=1$, find the intervals where the function $f\left(x\right)$ is monotonic and the equation of the tangent line at $x=1$.  
$(2)$ Let $g\left(x\right)=2f\left(x\right)-x^{2}-a^{2}$. If $g\left(x\right)\geqslant 0$ for all $x\geqslant 0$, determine the range of real numbers for $a$.

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