
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
Consider two points  $A(-1,\ 1), B(1,-1)$  on the coordinate plane. Let  $P$  be a point on the coordinate plane such that the absolute value of the  $x$ -coordinate 

of  $P$  is less than or equal to 1.  Draw the domain of the points  $P$  satisfying the condition (i) or (ii) as below, then find the area.

(i) There exists a parabola having the absolute value of the  $x$ -coordinate of the vertex which is more than or equal to 1 and passes through all points of  $A,\ P,\ B$ .

(ii) The points  $A,\ P,\ B$  are collinear.

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