
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  $k=1,2,\dots$ , let  $f_k$  be the number of times
\[\sin\left(\frac{k\pi x}{2}\right)\]
attains its maximum value on the interval  $x\in[0,1]$ . Compute
\[\lim_{k\rightarrow\infty}\frac{f_k}{k}.\]

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