
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
Apollonius, a famous mathematician in ancient Greece, conducted profound and systematic research on conic sections. One of his research results is the Apollonius circle, which refers to: given a moving point $M$ and two fixed points $Q$ and $P$, the ratio of the distances $\frac{|MQ|}{|MP|}=λ(λ＞0,λ≠1)$, then the locus of point $M$ is the Apollonius circle. Given that the locus of moving point $M$ is the Apollonius circle with the equation $x^{2}+y^{2}=\frac{1}{4}$, the fixed point $Q$ is a point on the $x$-axis, $P(-1,0)$, and $λ=\frac{1}{2}$, if point $B(1,2)$, then the minimum value of $\frac{1}{2}|MP|+|MB|$ is ______.

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