
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 vectors $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ in the same plane, where $\overrightarrow{a}=(-1,2)$,
(1) If $|\overrightarrow{c}|=\sqrt{5}$ and $\overrightarrow{a}$ is parallel to $\overrightarrow{c}$, find the coordinates of $\overrightarrow{c}$;
(2) If $|\overrightarrow{b}|=\frac{\sqrt{5}}{2}$ and $(\overrightarrow{a}+2\overrightarrow{b}) \perp (2\overrightarrow{a}-\overrightarrow{b})$, find $|2\overrightarrow{a}+\overrightarrow{b}|$.

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