
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 $\overrightarrow{a}=(2,\;4)$ and $\overrightarrow{b}=(-2,\;x)$.  
$(Ⅰ)$ When $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the value of $x$.  
$(Ⅱ)$ When $x=-1$, find the cosine value of the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$.  
$(Ⅲ)$ When $\overrightarrow{a}$ is perpendicular to $(4\overrightarrow{a}+\overrightarrow{b})$, find $|\overrightarrow{b}|$.

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