
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 $O_1: (x-4)^2 + y^2 = 4$, a moving point $P$ is on the line $x + \sqrt{3}y - b = 0$. Tangents to circles $O$ and $O_1$ at points $A$ and $B$ respectively are drawn from $P$. If there are exactly two points $P$ satisfying $PB = 2PA$, then the range of the real number $b$ is

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