
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]


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Let  $u$  be a real number. On the coordinate plane, consider two parabolas  $C_1: y=-x^2+1,\ C_2: y=(x-u)^2+u$ .
The range of  $u$  such that  $C_1$  and  $C_2$  have at least one point of intersection is expressed by  $a\leq u\leq b$  for some real numbers  $a,\ b$ .

(1) Find the values of  $a,\ b$ .

(2) When  $u$  satisfies  $a\leq u\leq b$ , let  $P_1(x_1,\ y_1),\ P_2(x_2,\ y_2)$  be the point of intersections of  $C_1$  and  $C_2$ .

Note : if the point of intersection is just one, then we regard the intersection as  $P_1=P_2$ .

Express  $2|x_1y_2-x_2y_1|$  in terms of  $u$ .

(3) Let  $f(u)$  be the expression obtained in (2), evaluate  $I=\int_ a^b f(u)du.$ 

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