
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
In the Cartesian coordinate system $xOy$, let $\overrightarrow{a}=\left(2\sin \theta ,1\right)$ and $\overrightarrow{b}=(1,\sin (\theta +\frac{π}{3}))$, where $\theta \in R$.<br/>$(1)$ If $\overrightarrow{a}\cdot \overrightarrow{b}=0$, find the value of $\tan \theta$;<br/>$(2)$ If $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, and $\theta \in (0,\frac{π}{2})$, find the value of $\theta$.

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