
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 $f(x)=2\sin(x+\frac{π}{3})\cdot\cos x$.  
$(1)$ If $f(x)=0$, find the set of values for $x$;  
$(2)$ If the graph of the function $f(x)$ is shifted to the right by $\frac{π}{3}$ units, and then all the points on the resulting graph have their y-coordinates unchanged and the x-coordinates stretched by a factor of $4$, find the monotonic increasing intervals of the function $g(x)$.

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