
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 $m=\left( \sin \left( x- \frac{\pi}{6} \right),1 \right)$, $n=(\cos x,1)$.  
$(1)$ If $m\parallel n$, find the value of $\tan x$;  
$(2)$ If the function $f(x)=m\cdot n$, $x\in[0,\pi]$, find the intervals of monotonic increase for $f(x)$.

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