
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 space vectors $\overrightarrow{a}=(1,0,1)$, $\overrightarrow{b}=(2,-1,0)$, $\overrightarrow{c}=(\lambda +4,-\lambda ,\lambda )$.
$(1)$ If $(\overrightarrow{a}+\overrightarrow{b}) \parallel \overrightarrow{c}$, find the value of $\lambda$;
$(2)$ If $k\overrightarrow{a}+\overrightarrow{b}$ is perpendicular to $2\overrightarrow{a}-\overrightarrow{b}$, find the value of $k$.

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