
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 that $a$ is a positive real number, the graph of the function $f(x) = ax^2 - a^2x - \frac{1}{a}$ intersects the x-axis at points A and B, with A to the left of B.

(1) Solve the inequality $f(x) > f(1)$ with respect to $x$;
(2) Find the minimum value of the distance AB;
(3) If $a \in [1, 2\sqrt{2}]$, determine the range of possible values for the length OA.

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