
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 two fixed points A(-2, 0) and B(1, 0), let point P be a moving point such that \(|PA| = 2|PB|\). If the curve C is the locus of point P and line l passes through the point (0, -3) intersecting curve C at two distinct points D(\(x_1\), \(y_1\)) and E(\(x_2\), \(y_2\)).

(Ⅰ) Find the equation of the locus of curve C;
(Ⅱ) Determine the range of the slope of the line l;
(Ⅲ) If \(x_1x_2 + y_1y_2 = 3\), find \(|DE|\).

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