
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 (1, 0), and a moving point P satisfying $|PA| = \sqrt{2}|PB|$.  
(Ⅰ) If the trajectory of point P is curve C, find the equation of curve C;  
(Ⅱ) Find the shortest distance from a point on the parabola $y^2 = x$ to the center of symmetry of curve C.

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