
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 $y=A\sin (\omega x+\varphi)$, where $x\in\mathbb{R}$, $A > 0$, $\omega > 0$, $|\varphi| < \frac{\pi}{2}$, if the coordinates of one of the highest points on the graph of the function are $\left(\frac{\pi}{6}, 3\right)$, and the coordinates of the symmetry center adjacent to it are $\left(-\frac{\pi}{12}, 0\right)$.

$(1)$ Find the expression of the function $y=A\sin (\omega x+\varphi)$;

$(2)$ Find the minimum value of the function, and write down the set of values of the independent variable $x$ when the function takes its minimum value.

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