
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 quadratic function $f(x) = ax^2 + x + 1$ (where $a \in \mathbb{R}, a \neq 0$),

1. If the solution set of the inequality $f(x) > 0$ is $(-\frac{1}{3}, \frac{1}{2})$, find the value of the real number $a$;
2. When $a \in [-2, 0]$, the inequality $f(x) > 0$ always holds, determine the range of real values that $x$ can take;
3. For $x \in [0, 2]$, if the inequality $f(x) > 0$ always holds, determine the range of real values that $a$ can take.

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