
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 $f(x)=\cos^2x+2\sqrt{3}\sin{x}\cos{x}-\sin^2x$.  
$(1)$ If $f\left(\frac{x_0}{2}\right)=\frac{1}{5}$, where $x_0\in[0,\frac{\pi}{3}]$, find the value of $\cos{2x_{0}}$.  
$(2)$ Let $A$, $B$, and $C$ be the three interior angles of $\triangle ABC$, and $f(B)=2$. If $D$ is a point on side $AC$ such that $AD=3DC=3$ and $\angle A=\angle ABD=\theta$, find the value of $\sin{\theta}$.

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