
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 plane vectors $\overrightarrow{a}=(3,4)$, $\overrightarrow{b}=(9,x)$, $\overrightarrow{c}=(4,y)$, $\overrightarrow{a}{\parallel}\overrightarrow{b}$, $\overrightarrow{a}{\perp}\overrightarrow{c}$.  
$(1)$ Find $\overrightarrow{b}$ and $\overrightarrow{c}$.  
$(2)$ If $\overrightarrow{m}=2\overrightarrow{a}-\overrightarrow{b}$, $\overrightarrow{n}=\overrightarrow{a}+\overrightarrow{c}$, find the angle between vector $\overrightarrow{m}$ and vector $\overrightarrow{n}$.

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