
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
The argument of the complex numbers \( z_{1}, z_{2}, z_{3} \) are \( \alpha, \beta, \gamma \) respectively, and \( \left|z_{1}\right|=1,\left|z_{2}\right|=k \), \( \left|z_{3}\right|=2-k \). Given that \( z_{1} + z_{2} + z_{3} = 0 \), determine the value of \( k \) for which \( \cos (\beta - \gamma) \) achieves its maximum or minimum value. Also, find the maximum and minimum values.

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