
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
Let the function $f(x)=A\sin (\omega x+\varphi)$ where $A > 0$, $\omega > 0$, and $|\varphi| < \frac{\pi}{2}$ have its highest point $D$ at coordinates $\left( \frac{\pi}{8},2\right)$. When moving from the highest point $D$ to the adjacent lowest point, the intersection point of the function graph with the $x$-axis is $\left( \frac{3\pi}{8},0\right)$;  
$(1)$ Find the analytical expression of the function $f(x)$.  
$(2)$ When $x\in\left[- \frac{\pi}{4}, \frac{\pi}{4}\right]$, find the maximum and minimum values of the function $f(x)$ and the corresponding values of the independent variable $x$ at which the maximum and minimum values are obtained, respectively.  
$(3)$ If the graph of the function $y=f(x)$ is translated to the right by $\frac{\pi}{4}$ units to obtain the graph of the function $y=g(x)$, find the intervals of decrease and the center of symmetry for the function $y=g(x)$.

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