
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 two vectors in a plane $\overrightarrow{a}=(1,2),\overrightarrow{b}=(-3,2)$.<br/>$(1)$ If $(k\overrightarrow{a}+2\overrightarrow{b})$<span class="special-symbol">∥</span>$(2\overrightarrow{a}-4\overrightarrow{b})$, find the real number $k$;<br/>$(2)$ If the vector $\overrightarrow{c}$ is a unit vector, and $|\overrightarrow{c}-\overrightarrow{b}|=2\sqrt{5}$, find the coordinates of $\overrightarrow{c}$.

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