
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 the parabola \( y = ax^2 + bx + c \) that passes through points \( A(-1,0) \) and \( B(2,0) \), for any nonzero real number \( a \), it never passes through point \( P(x_0 + 1, 2x_0^2 - 2) \). Determine the coordinates of point \( P \).

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