
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 $f(x)=2\cos \left(x- \frac{\pi}{3}\right)+2\sin \left( \frac{3\pi}{2}-x\right) +1$  
$(1)$ Find the center of symmetry of the function $f(x)$.  
  
$(2)$ If the line $y=a$ has no common points with the graph of the function $f(x)$, find the range of the real number $a$.  
$(3)$ If $f(x)= \frac{6}{5}$, find the value of $\cos \left(2x- \frac{\pi}{3}\right)$.

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