
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 Cartesian coordinate system $xOy$, given vectors $\overrightarrow{m}=( \frac { \sqrt {2}}{2},- \frac { \sqrt {2}}{2})$ and $\overrightarrow{n}=(\sin x,\cos x)$, where $x\in(0, \frac {π}{2})$.
$(1)$ If $\overrightarrow{m}\perp \overrightarrow{n}$, 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