
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 three vectors in the plane $\overrightarrow{a}=(3,2)$, $\overrightarrow{b}=(-1,2)$, $\overrightarrow{c}=(4,1)$:  
$(1)$ If $(\overrightarrow{a}+k\overrightarrow{c})\parallel(2\overrightarrow{b}+n\overrightarrow{c})$, find the real number $k$;  
$(2)$ Find the real numbers $m$, $n$ such that $\overrightarrow{a}=m\overrightarrow{b}-n\overrightarrow{c}$.

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