
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 \(O xy\), the coordinates of point \(A\) are given by the equation \(2 a^{2} - 2 a x - 6 a y + x^{2} + 2 x y + 5 y^{2} = 0\), and a parabola with vertex at point \(B\) is given by the equation \(a x^{2} + 4 a^{2} x - a y + 4 a^{3} + 2 = 0\). Find all values of the parameter \(a\) for which points \(A\) and \(B\) lie on the same side of the line \(x + y = 3\) (the points \(A\) and \(B\) do not lie on this line).

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