
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 the angle between the unit vectors $\overrightarrow{e_{1}}$ and $\overrightarrow{e_{2}}$ is ${120}^{{}^\circ }$, find the value of $\overrightarrow{e_{1}} \cdot \overrightarrow{e_{2}}$. Additionally, determine the minimum value of $| \overrightarrow{e_{1}} - \lambda \overrightarrow{e_{2}} |$ where $\lambda \in \mathbb{R}$.

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