
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 \(Oxy\), the coordinates of point \(A\) are given by the equation:
\[ 5a^2 - 4ay + 8x^2 - 4xy + y^2 + 12ax = 0 \]
and the parabola with its vertex at point \(B\) is given by the equation:
\[ ax^2 - 2a^2 x - ay + a^3 + 3 = 0 \]
Find all values of the parameter \(a\) for which points \(A\) and \(B\) lie on the same side of the line \(2x - y = 5\) (points \(A\) and \(B\) do not lie on this line).

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