
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
On the coordinate plane, let  $C$  be a circle centered  $P(0,\ 1)$  with radius 1. let  $a$  be a real number  $a$  satisfying  $0<a<1$ . Denote by  $Q,\ R$  intersection points of the line  $y=a(x+1) $  and  $C$ .

(1) Find the area  $S(a)$  of  $\triangle{PQR}$ .

(2) When  $a$  moves in the range of  $0<a<1$ , find the value of  $a$  for which  $S(a)$  is maximized.

*2011 Tokyo University entrance exam/Science, Problem 1*

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