
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 the vectors $\overrightarrow{a}=(\sin \theta, -\frac{2}{5})$ and $\overrightarrow{b}=(1, 2\cos \theta)$,
(1) If $\overrightarrow{a}$ and $\overrightarrow{b}$ are perpendicular to each other, find the value of $\tan \theta$;
(2) If $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the value of $\cos (\frac{\pi}{2}+2\theta)$.

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