
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 $f(x)=\sin (2\omega x+ \frac {\pi}{3})+ \frac { \sqrt {3}}{2}+a$ ($\omega > 0$), and the graph of $f(x)$ has its first highest point on the right side of the $y$-axis at the $x$-coordinate of $\frac {\pi}{6}$.  
$(1)$ Find the value of $\omega$;  
$(2)$ If the minimum value of $f(x)$ in the interval $\left[- \frac {\pi}{3}, \frac {5\pi}{6}\right]$ is $\sqrt {3}$, find the value of $a$;  
$(3)$ If $g(x)=f(x)-a$, then how can the graph of $g(x)$ be obtained from the graph of $y=\sin x$ ($x\in\mathbb{R}$) through transformations? Also, write down the axis of symmetry and the center of symmetry for $g(x)$.

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