
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 vectors $\overrightarrow{a}=(1,\sin \theta)$ and $\overrightarrow{b}=(3,1)$,
$(1)$ When $\theta= \frac{\pi}{6}$, find the coordinates of the vector $2 \overrightarrow{a}+ \overrightarrow{b}$;
$(2)$ If $\overrightarrow{a} \parallel \overrightarrow{b}$, and $\theta \in (0, \frac{\pi}{2})$, find the value of $\sin (2\theta+ \frac{\pi}{4})$.

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