
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 $\overrightarrow{a}=(-3,0)$, $\overrightarrow{b}=(\mu+3,-1)$, $\overrightarrow{c}=(1,\lambda)$.  
$(1)$ If $\lambda =8$ and $\mu =-6$, find the angle between the vectors $\overrightarrow{a}-\overrightarrow{c}$ and $\overrightarrow{b}$.  
$(2)$ If $(\overrightarrow{a}+\overrightarrow{b})\perp\overrightarrow{c}$ and the magnitude of the projection of $\overrightarrow{a}$ onto $\overrightarrow{c}$ is $1$, find the values of $\lambda$ and $\mu$.

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