
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 $f(x)=2\sin(2x+\frac{π}{4})$.  
$(1)$ Find the smallest positive period and the interval of monotonicity of $f(x)$;  
$(2)$ Find the maximum and minimum values of $f(x)$ on $[-\frac{π}{8},\frac{π}{2}]$, and determine the corresponding values of $x$;  
$(3)$ If the function $g(x)=f(x)-\sqrt{2}$ has exactly $3$ zeros in the interval $[-\frac{π}{8},m]$, determine the range of values for $m$.

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