
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 $\overrightarrow{a}=(x,4,1),\overrightarrow{b}=(-2,y,-1),\overrightarrow{c}=(3,-2,z),\overrightarrow{a}∥\overrightarrow{b},\overrightarrow{b}⊥\overrightarrow{c}$, find:<br/>$(1)$ the value of $x+y+z$;<br/>$(2)$ 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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