
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 circles O: $x^{2}+y^{2}=1$ and $O_{1}: (x-4)^{2}+y^{2}=4$, there is a moving point P on the line $x+\sqrt{3}y+b=0$. Tangent lines from P to circles O and $O_{1}$ respectively meet the circles at points A and B. If there are exactly two points P such that $PB=2PA$, then the range of values for the real number b is ______.

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