
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 $f(x)=\sin(\frac{π}{3}-2x)$.  
$(1)$ Find the monotonic intervals of $f(x)$;  
$(2)$ Given $g(x)=x^{2}-2x+m-3$, for all $x_1\in[\frac{π}{12},\frac{π}{2}]$, there exists $x_{2}\in[-2,m]$, such that $f(x_{1})=g(x_{2})$ holds true. Find the range of values for $m$.

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