
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 $f(x) = (x^2 - x - \frac{1}{a})e^{ax}$ ($a \neq 0$).

(Ⅰ) When $a= \frac{1}{2}$, find the zeros of the function $f(x)$.

(Ⅱ) Determine the intervals of monotonicity for $f(x)$.

(Ⅲ) When $a > 0$, if $f(x) + \frac{2}{a} \geq 0$ holds for all $x \in \mathbb{R}$, find the range of $a$.

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