
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
In the plane rectangular coordinate system $xOy$, given vectors $\overrightarrow{m}=(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$, $\overrightarrow{n}=(\sin x, \cos x)$, where $x\in (0, \frac{π}{2})$.  
$(1)$ If $\overrightarrow{m}\bot \overrightarrow{n}$, find the value of $\tan x$;  
$(2)$ If the angle between $\overrightarrow{m}$ and $\overrightarrow{n}$ is $\frac{π}{3}$, find the value of $x$.

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