
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
Given points $A(-1,0)$, $B(2,0)$, and circle $C:{(x-2)^2}+{(y-m)^2}=\frac{1}{4}(m > 0)$, find the range of real number $m$ such that there exists a point $P$ on the circle for which $|PA|=2|PB|$.

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