
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 $\overrightarrow{{e_1}},\overrightarrow{{e_2}}$ be a basis in a plane, if $\overrightarrow{AB}=\overrightarrow{{e_1}}-k\overrightarrow{{e_2}}$, $\overrightarrow{CB}=4\overrightarrow{{e_1}}-2\overrightarrow{{e_2}}$, and $\overrightarrow{CD}=3\overrightarrow{{e_1}}-3\overrightarrow{{e_2}}$ and points $A$, $B$, and $D$ are collinear, determine the value of $k$.

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