
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 $f(x)=2\sin^2\left(\frac{\pi}{4}+x\right)-\sqrt{3}\cos2x$.
$(1)$ Find the interval where $f(x)$ is monotonically decreasing.
$(2)$ If the equation $f(x)-m=2$ has a unique solution for $x\in\left[\frac{\pi}{4},\frac{\pi}{2}\right]$, find the range of real numbers for $m$.

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