
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
In a rectangular coordinate system, the ellipse \(\Omega: \frac{x^{2}}{4}+y^{2}=1\) has a moving point \(P\) on \(\Omega\), and \(A\) and \(B\) are two fixed points with \(B\) at coordinates \((0,3)\). If the minimum area of triangle \(PAB\) is 1 and the maximum area is 5, what is the length of segment \(AB\)?

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