
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 the Cartesian coordinate system $xOy$, given the circle $O: x^2 + y^2 = 1$ and the circle $C: (x-4)^2 + y^2 = 4$, a moving point $P$ is located between two points $E$ and $F$ on the line $x + \sqrt{3}y - 2 = 0$. Tangents to circles $O$ and $C$ are drawn from point $P$, with the points of tangency being $A$ and $B$, respectively. If it satisfies $PB \geqslant 2PA$, then the length of the segment $EF$ is.

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