
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 $f(x) = (ax^2 + bx + c)e^x$ and it is known that $f(0) = 1$ and $f(1) = 0$.
(I) If $f(x)$ is monotonically decreasing in the interval $[0, 1]$, find the range of the real number $a$.
(II) When $a = 0$, does there exist a real number $m$ such that the inequality $2f(x) + 4xe^x \geq mx + 1 \geq -x^2 + 4x + 1$ for any $x \in \mathbb{R}$? If it exists, find the value of $m$; if not, explain why.

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