
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 $f(x)=2\sin(\omega x+\varphi)-1$ where $(0 < \omega < 3, 0 < \varphi < \frac{π}{2})$, satisfies ______.<br/>$(1)$ Find the analytical expression of $f(x)$ and write down the interval where $f(x)$ is monotonically decreasing;<br/>$(2)$ Shift the graph of $y=f(x)$ to the right by $\frac{π}{6}$ units, then up by $\frac{1}{2}$ unit, resulting in the graph of the function $y=g(x)$. If the maximum value of $g(x)$ on the interval $[-\frac{π}{3}, m]$ is $\frac{3}{2}$, find the minimum value of the real number $m$.<br/>From the given conditions: ① one zero of the function $f(x)$ is at $0$; ② the distance between adjacent axes of symmetry on the graph of $f(x)$ is $\frac{π}{2}$; ③ the coordinates of a lowest point on the graph of $f(x)$ are $({\frac{{2π}}{3},-2})$, choose any two of these three conditions and incorporate them into the above question, and provide the solution to the problem.<br/>Note: If different combinations are chosen for answering separately, scoring will be based on the first answer.

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