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

(1) If $(\overrightarrow{a} + \overrightarrow{b}) \parallel \overrightarrow{c}$, find the value of $x$;

(2) If $\overrightarrow{a} \cdot \overrightarrow{b} = \frac{1}{2}$, find the value of the function $\sin \left(x + \frac{\pi}{6}\right)$.

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