
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 $\overrightarrow m=({\sin(x-\frac{π}{6}),1})$ and $\overrightarrow n=(\cos x,1)$.

$(1)$ If $\overrightarrow{m}∥\overrightarrow{n}$ and $x\in \left(0,\pi \right)$, find the value of $x$.

$(2)$ Let $f(x)=\overrightarrow m•\overrightarrow n$, and $x\in \left[0,\pi \right]$. Find the interval where $f(x)$ is monotonically increasing.

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