
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
Let $f(x)=\cos(2x+\frac{π}{3})+2\cos^{2}x-1, x∈R$<br/>$(1)$ If the graph of the function $f(x)$ is shifted to the right by $\frac{π}{6}$ units, resulting in the graph of the function $y=g(x)$, find the interval of monotonic increase for the function $g(x)$;<br/>$(2)$ Find the solutions of the equation $f(x)+\frac{\sqrt{3}}{2}=0$ on the interval $\left[0,\pi \right]$.

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