
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 \( x^2 = 4y \) where \( y_1 + y_2 = 2 \) and \( y_1 \neq y_2 \), and the perpendicular bisector of line segment \( AB \) intersects the y-axis at point \( C \), find the maximum value of the area of \( \triangle ABC \).

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