
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 functions $f(x)=e^{x}(2x-1)$ and $g(x)=ax-a$ $(a\in \mathbb{R})$.

$(1)$ If $y=g(x)$ is a tangent line of the curve $y=f(x)$, find the value of $a$;

$(2)$ Given $a < 1$, if there exists a unique integer $x_{0}$ such that $f(x_{0}) < g(x_{0})$, find the range of values for $a$.

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