
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
A student used the "five-point method" to draw the graph of the function $f(x)=A\sin(\omega x+\varphi)$ ($\omega\ \gt 0$, $|\varphi|＜\frac{π}{2}$) within one period. The student listed and filled in some of the data in the table below:

| $\omega x+\varphi$ | $0$ | $\frac{π}{2}$ | $\pi$ | $\frac{{3π}}{2}$ | $2\pi$ |
|-------------------|-----|---------------|-------|------------------|-------|
| $x$               |     |               | $\frac{{3π}}{8}$ | $\frac{{5π}}{8}$ |       |
| $A\sin(\omega x+\varphi)$ | $0$ | $2$ |       | $-2$ | $0$ |

$(1)$ Please complete the data in the table and write the analytical expression of the function $f(x)$ on the answer sheet.

$(2)$ Move the graph of $f(x)$ to the left by $\theta$ units to obtain the graph of $g(x)$. If the graph of $g(x)$ is symmetric about the line $x=\frac{π}{3}$, find the minimum value of $\theta$.

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