
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 $26a^2 - 22ax - 20ay + 5x^2 + 8xy + 4y^2 = 0$, and the equation $ax^2 + 2a^2x - ay + a^3 + 1 = 0$ represents a parabola with its vertex at point $B$. Find all values of the parameter $a$ for which points $A$ and $B$ lie on opposite sides of the line $3x - y = 4$ (points $A$ and $B$ do not lie on this line).

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