
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 vectors $a=(\sin x,\cos (\pi-x))$ and $b=(2\cos x,2\cos x)$, and the function $f(x)=a\cdot b+1$.

$(1)$ Find the center of symmetry of $f(x)$;

$(2)$ Find the maximum and minimum values of the function $f(x)$ in the interval $\left[0,\frac{\pi}{2}\right]$ and the corresponding values of $x$.

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