
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
Given the function $f(x)=2\sin^2\left(\frac{\pi}{4}+x\right)-\sqrt{3}\cos 2x$.
$(1)$ Find the smallest positive period of $f(x)$;
$(2)$ Find the maximum and minimum values of $f(x)$ when $x\in\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$;
$(3)$ If the inequality $|f(x)-m| < 2$ always holds for $x\in\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$, find the range of the real number $m$.

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