
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
Let  $a$  be a positive real numbers. Let  $t,\ u\ (t<u)$  be the  $x$  coordinates of the point of intersections of the curves :  $C_1:y=|\cos x|\ (0\leq x\leq \pi),\ C_2:y=a\sin x\ (0\leq x\leq \pi).$  Denote by  $S_1$  the area of the part bounded by  $C_1,\ C_2$  and  $y$ -axis in  $0\leq x\leq t$ , and by  $S_2$  the area of the part bounded by  $C_1,\ C_2$  in  $t\leq x\leq u$ . When  $a$  moves over all positive real numbers, find the minimum value of  $S_1+S_2$ .

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