
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
The famous Greek mathematician Apollonius discovered: "The locus of points in a plane whose distance ratio to two fixed points $A$ and $B$ is a constant value $\lambda (\lambda \neq 1)$ is a circle." Later, this circle was named after him and called the Apollonius circle. In the Cartesian coordinate system $xOy$ plane, with $A(2,0)$, $B(8,0)$, and $\frac{{|PA|}}{{|PB|}}=\frac{1}{2}$, the equation of the locus of point $P$ is ____.

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