
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 a parabola  $C : y=1-x^2$  in  $xy$ -palne with the origin  $O$ . Take two points  $P(p,\ 1-p^2),\ Q(q,\ 1-q^2)\ (p<q)$  on  $C$ .

(1) Express the area  $S$  of the part enclosed by two segments  $OP,\ OQ$  and the parabalola  $C$  in terms of  $p,\ q$ .

(2) If  $q=p+1$ , then find the minimum value of  $S$ .

(3) If  $pq=-1$ , then  find the minimum value of  $S$ .

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