
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 \(Oxy\), there is a point \(A\) with coordinates \((x; y)\) that satisfy the equation \(5a^{2} - 4ax + 6ay + x^{2} - 2xy + 2y^{2} = 0\), and a circle centered at point \(B\) given by the equation \(a^{2} x^{2} + a^{2} y^{2} - 4a^{3} x - 2ax + 2a^{2} y + 4a^{4} + 1 = 0\). Find all values of the parameter \(a\) for which points \(A\) and \(B\) lie on opposite sides of the line \(x = 3\) (points \(A\) and \(B\) do not lie on this line).

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