
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 $\overrightarrow{a}=(3,5,-4)$ and $\overrightarrow{b}=(2,1,8)$, compute $3\overrightarrow{a}-2\overrightarrow{b}$, the dot product of $\overrightarrow{a}$ and $\overrightarrow{b}$, and determine the relationship between $\lambda$ and $\mu$ such that $\lambda\overrightarrow{a}+\mu\overrightarrow{b}$ is perpendicular to the $z$-axis.

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