
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 vectors $\overrightarrow{a}=(\sin x, \sqrt {3}\cos x), \overrightarrow{b}=(-1,1), \overrightarrow{c}=(1,1)$, where $x\in[0,\pi]$.

$(1)$ If $(\overrightarrow{a}+ \overrightarrow{b})\parallel \overrightarrow{c}$, find the value of the real number $x$;  
$(2)$ If $\overrightarrow{a}\cdot \overrightarrow{b}= \frac {1}{2}$, find the value of the function $\sin (x+ \frac {\pi}{6})$.

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