
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
For two points $A$ and $B$ on the plane, if $\frac{|PA|}{|PB|}=λ(λ≠1)$, the locus of the point $P$ is a circle, which was first discovered by the ancient Greek mathematician Apollonius and named Apollonius circle. Given $A(1,0)$, $B(4,0)$, $D(0,3)$, if the moving point $P$ satisfies $\frac{|PA|}{|PB|}=\frac{1}{2}$, then the minimum value of $2|PD|+|PB|$ is ______.

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