
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
Does there exist a translation such that the parabola \( y = -x^2 \) after translation passes through the origin, and the area of the triangle formed by the vertex of the translated parabola and its two intersection points with the \( x \)-axis is 1? If it exists, find the equation of the function; if not, explain why.

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