
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: $\overrightarrow{a}=\left(x,4,1\right)$, $\overrightarrow{b}=\left(-2,y,-1\right)$, $\overrightarrow{c}=\left(3,-2,z\right)$, $\overrightarrow{a}$∥$\overrightarrow{b}$, $\overrightarrow{b}$⊥$\overrightarrow{c}$, find:  
$(1)$ $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$;  
$(2)$ The cosine value of the angle between $\overrightarrow{a}+\overrightarrow{c}$ and $\overrightarrow{b}+\overrightarrow{c}$.

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