
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 $\overrightarrow{{e}_{1}}$ and $\overrightarrow{{e}_{2}}$ be two non-collinear vectors in space. Given that $\overrightarrow{AB}=\overrightarrow{{e}_{1}}+k\overrightarrow{{e}_{2}}$, $\overrightarrow{BC}=5\overrightarrow{{e}_{1}}+4\overrightarrow{{e}_{2}}$, $\overrightarrow{DC}=-\overrightarrow{{e}_{1}}-2\overrightarrow{{e}_{2}}$, and points $A$, $B$, $D$ are collinear, find the real number $k=$____.

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