
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 \( z_{1} = \sin \alpha + 2i \) and \( z_{2} = 1 + i \cos \alpha \) (where \( \alpha \) is a real number and \( i \) is the imaginary unit), what is the minimum value of \( \frac{13 - \left| z_{1} + i z_{2} \right|^{2}}{\left| z_{1} - i z_{2} \right|} \)?

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