
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).

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

(Ⅱ) If $a=-1$, the equation $f(x) = k \cdot g(x)$ has exactly one root, find the range of the real number $k$.

(Ⅲ) 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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