
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
Let \( a, b \in \mathbb{R} \). The equation \( \left(x^{2} - ax + 1\right)\left(x^{2} - bx + 1\right) = 0 \) has 4 real roots that form a geometric sequence with common ratio \( q \). If \( q \in \left[ \frac{1}{3}, 2 \right] \), determine the range of the value of \( ab \).

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