
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 a parabola \( y^{2}=6x \) with two moving points \( A\left(x_{1}, y_{1}\right) \) and \( B\left(x_{2}, y_{2}\right) \) on it, where \( x_{1} \neq x_{2} \) and \( x_{1} + x_{2} = 4 \). The perpendicular bisector of line segment \( \mathrm{AB} \) intersects the \( x \)-axis at point \( C \). Find the maximum area of triangle \( \mathrm{ABC} \).

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