
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 $xOy$, we are given Circle $O: x^{2}+y^{2}=1$ and Circle $O_{1}: (x-4)^{2}+y^{2}=4$. A moving point $P$ lies on the line $x + \sqrt{3}y - b = 0$. Tangents to Circle $O$ and Circle $O_{1}$ at points $A$ and $B$, respectively, pass through point $P$. If there are exactly two such points $P$ that satisfy $PB = 2PA$, then the range of the real number $b$ is \_\_\_\_\_\_.

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