
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 $f(x)=e^{x}-ax^{2}-bx-1$, where $a$, $b\in\mathbb{R}$, and $e$ is the base of the natural logarithm. It is known that $|x-a|\geqslant f(x)$ always holds. Find the range of values for the real number $a$.  
$(1)$ If the tangent line of the function $f(x)$ at the point $(1,f(1))$ is $y=(e-1)x-1$, find the values of the real numbers $a$ and $b$;  
$(2)$ Let $g(x)$ be the derivative of the function $f(x)$. Find the minimum value of the function $g(x)$ in the interval $[0,1]$.

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