
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 $f(x)=x^{2}+ax+1$, $g(x)=e^{x}$, where $e$ is the base of the natural logarithm.

(1) If $a=1$, find the maximum value of the function $y=f(x) \cdot g(x)$ in the interval $[-2,0]$;

(2) If $a=-1$, the equation $f(x)=k \cdot g(x)$ has exactly one root with respect to $x$, find the range of the real number $k$;

(3) If for any $x_{1}, x_{2} \in [0,2]$, $x_{1} \neq x_{2}$, the inequality $|f(x_{1})-f(x_{2})| < |g(x_{1})-g(x_{2})|$ always holds, find the range of the real number $a$.

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