
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
In the plane \( Oxy \), point \( A \) with coordinates \( (x, y) \) satisfies the equation \( 2a^2 + 2ax + x^2 - 2xy + 2y^2 = 0 \), and a circle centered at point \( B \) is given by the equation \( a^2 x^2 + a^2 y^2 - 2a^3 x - 6a x - 2a^2 y + a^4 + 9 = 0 \). Find all values of parameter \( a \) for which points \( A \) and \( B \) lie on opposite sides of the line \( x = 4 \) (points \( A \) and \( B \) do not lie on this line).

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