
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
Consider the coordinate plane \( Oxy \) where point \( A \) has coordinates \((x, y)\) that satisfy the equation \( 5a^2 - 6ax - 4ay + 2x^2 + 2xy + y^2 = 0 \), and a circle centered at point \( B \) given by the equation \( a^2x^2 + a^2y^2 - 6a^2x - 2a^3y + 4ay + a^4 + 4 = 0 \). Find all values of the parameter \( a \) such that points \( A \) and \( B \) lie on opposite sides of the line \( y = 1 \) (points \( A \) and \( B \) do not lie on this line).

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