
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 that $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$, the parabola intersects the x-axis at points A and B, and intersects the y-axis at point C. The vertex of the parabola lies on the line $y = -1$. If $\triangle ABC$ is a right-angled triangle, determine the maximum area of $\triangle ABC$.

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