
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 positive integer  $n$ , let  $S_n$  be the total sum of the intervals of  $x$  such that  $\sin 4n x\geq \sin x$  in  $0\leq x\leq \frac{\pi}{2}.$ Find  $\lim_{n\to\infty} S_n.$ 60 point

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