
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 $f(x) = ax^{2}e^{x}$, where $a \in \mathbb{R}$ and $e$ is the base of the natural logarithm.

(Ⅰ) If the slope of the tangent line to the function $f(x)$ at the point $(1, f(1))$ is $6e$, find the value of the real number $a$.

(Ⅱ) When $a \neq 0$, discuss the monotonicity of the function $f(x)$.

(Ⅲ) If the inequality $f(x) + xe^{x} + 1 \geq e^{x}$ holds for all $x$ in the interval $(-\infty, 0]$, find the range of values for the real number $a$.

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