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

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