
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  $ f(x)\equal{}\sin 3x\plus{}\cos x,\ g(x)\equal{}\cos 3x\plus{}\sin x.$ 

(1) Evaluate  $ \int_0^{2\pi} \{f(x)^2\plus{}g(x)^2\}\ dx$ .

(2) Find the area of the region bounded by two curves  $ y\equal{}f(x)$  and  $ y\equal{}g(x)\ (0\leq x\leq \pi).$ 

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