
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 the complex numbers \( z_1 = \frac{\cos x^2}{\sin y^2} + i \cdot \frac{\cos y^2}{\sin x^2} \) and \( z_2 = x + yi \) for \( x, y \in \mathbb{R} \). If \( |z_1| \) is always equal to \( \sqrt{2} \) for all \( x, y \) in the interval \(\left[-\sqrt{\frac{\pi}{2}}, \sqrt{\frac{\pi}{2}}\right] \), find the maximum and minimum values of \( |z_1 - z_2| \).

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