
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 \( x, y \in \left[-\sqrt{\frac{\pi}{2}}, \sqrt{\frac{\pi}{2}}\right] \), \( z_{1} = \frac{\cos x^{2}}{\sin y^{2}} + \frac{\cos y^{2}}{\sin x^{2}} i \), and \( \left|z_{1}\right| = \sqrt{2} \). If \( z_{2} = x + y i \), what is the range of values for \( \left|z_{2} - z_{1}\right| \)?

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