
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 $f(x)=\sin^4x-\cos^4x+2\sqrt{3}\sin x \cos x-1$.  
$(1)$ Find the smallest positive period of $f(x)$ and the center of symmetry of its graph.  
$(2)$ When $x\in[-\frac{\pi}{6},\frac{2\pi}{3}]$, the maximum value of the function $g(x)=af(x)+b$ is $11$, and the minimum value is $3$. Find the values of the real numbers $a$ and $b$.

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