
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 $f(x)=\sin x\cos x+\sqrt{3}\cos^2x-\frac{\sqrt{3}}{2}$, $g(x)=f(x-\theta)$, $0 < \theta < \frac{\pi}{2}$, and the graph of $g(x)$ is symmetric about the point $(\frac{\pi}{6},0)$.
$(1)$ Find $\theta$.
$(2)$ Let $\triangle ABC$ have sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$, respectively. The circumradius is $R$, and $g(\frac{A}{8})=-\frac{1}{2}$, $b=1$, $R=\frac{\sqrt{21}}{3}$. If point $D$ is the trisection point of side $BC$ closer to $B$, find the length of $AD$.

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