
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 $f(x) = x^2 + ax + 1$, where $a \in \mathbb{R}$, and $g(x) = e^x$ (where $e$ is the base of natural logarithms):
(1) Let the function $H(x) = \frac{f(x)}{g(x)}$, find the intervals of monotonicity for $H(x)$;
(2) If for any $x_1, x_2 \in [0, 2]$ with $x_1 > x_2$, it holds that $|f(x_1) - f(x_2)| < |g(x_1) - g(x_2)|$, find the range of the real number $a$.

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