
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
For  $0<k<2$ , consider two curves  $C_1: y=\sin 2x\ (0\leq x\leq \pi),\ C_2: y=k\cos x\ (0\leqq x\leqq \pi).$ Denote by  $S(k)$  the sum of the areas of four parts enclosed by  $C_1,\ C_2$  and two lines  $x=0,\ x=\pi$ .
Find the minimum value of  $S(k).$ *2010 Nagoya Institute of Technology entrance exam*

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