
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 $\overrightarrow{a}=(1,1),\overrightarrow{b}=(4,x),\overrightarrow{u}=\overrightarrow{a}+2\overrightarrow{b},\overrightarrow{v}=2\overrightarrow{a}+\overrightarrow{b}$.  
$(1)$ If $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the value of real number $x$.  
$(2)$ If $(\overrightarrow{u}-2\overrightarrow{v})$ is perpendicular to $(\overrightarrow{u}+\overrightarrow{v})$, find the value of real number $x$.

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