
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 $f(x)=e^{x}-2(a-1)x-b$, where $e$ is the base of the natural logarithm.
$(1)$ If the function $f(x)$ is monotonic on the interval $[0,1]$, find the range of values for the real number $a$;
$(2)$ Given the function $g(x)=e^{x}-(a-1)x^{2}-bx-1$, and $g(1)=0$, if the function $g(x)$ has exactly $3$ zeros on the interval $[0,1]$, find the range of values for the real number $a$.

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