
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 $f\left(x\right)=e^{x}-ax+\frac{1}{2}{x}^{2}$, where $a \gt -1$.<br/>$(Ⅰ)$ When $a=0$, find the equation of the tangent line to the curve $y=f\left(x\right)$ at the point $\left(0,f\left(0\right)\right)$;<br/>$(Ⅱ)$ When $a=1$, find the extreme values of the function $f\left(x\right)$;<br/>$(Ⅲ)$ If $f(x)≥\frac{1}{2}x^2+x+b$ holds for all $x\in R$, find the maximum value of $b-a$.

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