
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 - 4x + 2, and the function g(x) = (1/3)^f(x).
(I) If the function f(x) has opposite monotonicity on the intervals (-∞, 2] and [2, +∞), find the analytical expression of f(x);
(II) If a < 0, find the range of a such that the inequality g(x) ≤ 9 holds for all x in the interval (0, 1/2];
(III) Given that a ≤ 1, find the range of the real number a such that the equation y = f(x) - log₂(x/8) has exactly one root in the interval [1, 2].

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