
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 $A(-1,0)$ and $B(1,0)$, and a moving point $P$ that satisfies $|PA|= \sqrt {2}|PB|$.
(I) If the trajectory of point $P$ forms a curve $C$, derive the equation of curve $C$.
(II) Find the shortest distance from the center of symmetry of curve $C$ to a point on the parabola $y^{2}=x$.

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