
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 $f(x)=A\sin (\omega x+\varphi)$ where $A > 0, \omega > 0, 0 < \varphi < \frac{\pi}{2}$, the graph intersects the $x$-axis at points such that the distance between two adjacent points is $\frac{\pi}{2}$. Also, a lowest point on the graph is $M\left( \frac{2\pi}{3},-2 \right)$.

$(1)$ Find the expression for $f(x)$;

$(2)$ Find the intervals where $f(x)$ is monotonically decreasing;

$(3)$ When $x\in\left[ \frac{\pi}{12}, \frac{\pi}{2}\right]$, find the range of $f(x)$.

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