
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 $\overrightarrow{AB}=\left(-1,3\right)$, $\overrightarrow{BC}=\left(3,m\right)$, $\overrightarrow{CD}=\left(1,n\right)$, and $\overrightarrow{AD}$ is parallel to $\overrightarrow{BC}$.  
$(1)$ Find the value of real number $n$.  
$(2)$ If $\overrightarrow{AC}$ is perpendicular to $\overrightarrow{BD}$, find the value of real number $m$.

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