
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 $k$ is a real number, $f(x)=2\sin^{2}(\frac{\pi}{4}+x)-k\cdot\cos2x$.  
$(1)$ If $k=0$, find the solutions of the equation $f(x)=1$ on the interval $\left[0,\pi \right]$;  
$(2)$ If $k=\sqrt{3}$, find the interval of monotonic decrease for the function $y=f(x)$, $x\in \mathbb{R}$;  
$(3)$ Given $a$ is a real number and $k=\sqrt{3}$, if the inequality $|f(x)-a| \lt 2$ holds for $x\in[\frac{\pi}{4},\frac{\pi}{2}]$ always, find the range of values for $a$.

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