
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
Given two moving points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) on the parabola \(y^2 = 6x\), where \(x_1 \neq x_2\) and \(x_1 + x_2 = 4\). If the perpendicular bisector of segment \(AB\) intersects the \(x\)-axis at point \(C\), find the maximum area of triangle \(ABC\).

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