
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 two unit vectors $\overrightarrow{e_1}$ and $\overrightarrow{e_2}$ with an angle of $\dfrac{2\pi}{3}$ between them, $\overrightarrow{a}= \overrightarrow{e_1}-2\overrightarrow{e_2}$, $\overrightarrow{b}=k\overrightarrow{e_1}+\overrightarrow{e_2}$. If $\overrightarrow{a} \cdot \overrightarrow{b}=0$, then the value of the real number $k$ is ______.

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