
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 a function $f(x)=Asin(2ωx+φ)$ where $A>0$, $ω>0$, and $|φ|≤\frac{π}{2}$, with an amplitude of $2$ and an initial phase of $\frac{π}{6}$.<br/>$(1)$ If the distance between adjacent axes of symmetry of the function $f(x)$ is $\frac{π}{2}$,<br/>① Find the value of $ω$ and the interval where $f(x)$ is monotonically decreasing;<br/>② Find the extreme values of $f(x)$ on the interval $[0$,$\frac{π}{2}]$, and the corresponding values of $x$.<br/>$(2)$ If the function $g(x)=f(x)-\sqrt{3}$ has exactly $5$ zeros in the interval $\left[0,3\pi \right]$, determine the range of values for $ω$.

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