
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 $f(x)=A\sin \left(x+\frac{\pi }{3}\right)$, where $x\in \mathbb{R}$, and $f\left(\frac{5\pi }{12}\right)=\frac{3\sqrt{2}}{2}$.

$(1)$ Find the value of $A$;

$(2)$ If $f(\theta )-f(-\theta )=\sqrt{3}$, where $\theta \in \left(0,\frac{\pi }{2}\right)$, find $f\left(\frac{\pi }{6}-\theta \right)$.

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