
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 that $\overset{\to }{{{{e}\_{1}}}}\,$ and $\overset{\to }{{{{e}\_{2}}}}\,$ are two unit vectors with an angle of $\dfrac{2}{3}\pi$ between them, $\overset{\to }{{a}}\,$ = $\overset{\to }{{{{e}\_{1}}}}\,$ - $2\overset{\to }{{{{e}\_{2}}}}\,$, and $\overset{\to }{{b}}\,$ = $k\overset{\to }{{{{e}\_{1}}}}\,$ + $\overset{\to }{{{{e}\_{2}}}}\,$. If $\overset{\to }{{a}}\,$ $\cdot$ $\overset{\to }{{b}}\,$ = 0, find the value of $k$.

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