
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 a real number  $x$ , let  $f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt$ .

(1) Find the minimum value of  $f(x)$ .

(2) Evaluate  $\int_0^1 f(x)\ dx$ .

*2011 Tokyo Institute of Technology entrance exam, Problem 2*

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