
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 $f(x)=\sin(2x+\frac{π}{6})+2\sin^2x$.  
$(1)$ Find the center of symmetry and the interval of monotonic decrease of the function $f(x)$;  
$(2)$ If the graph of $f(x)$ is shifted to the right by $\frac{π}{12}$ units, resulting in the graph of the function $g(x)$, find the maximum and minimum values of the function $g(x)$ on the interval $[0,\frac{π}{2}]$.

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