
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 points $A(-2,4)$, $B(3,-1)$, $C(-3,-4)$. Let $\overrightarrow{AB} = \overrightarrow{a}$, $\overrightarrow{BC} = \overrightarrow{b}$, $\overrightarrow{CA} = \overrightarrow{c}$, and $\overrightarrow{CM} = 3\overrightarrow{c}$, $\overrightarrow{CN} = -2\overrightarrow{b}$.

$(1)$ Find $3\overrightarrow{a} + \overrightarrow{b} - 3\overrightarrow{c}$;

$(2)$ Find the real numbers $m$, $n$ such that $\overrightarrow{a} = m\overrightarrow{b} + n\overrightarrow{c}$;

$(3)$ Find the coordinates of $M$, $N$ and the coordinates of vector $\overrightarrow{MN}$.

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