
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
Let $A$, $B$, $C$, $D$ be four points in a plane, with $A(1,3)$, $B(2,-2)$, $C(4,1)$.  
$(1)$ If $\overrightarrow{AB}= \overrightarrow{CD}$, find the coordinates of point $D$;  
$(2)$ Let vector $\overrightarrow{a}= \overrightarrow{AB}$, $\overrightarrow{b}= \overrightarrow{BC}$. If $k \overrightarrow{a}- \overrightarrow{b}$ is parallel to $\overrightarrow{a}+3 \overrightarrow{b}$, find the value of the real number $k$.

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