
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
Let the complex numbers
$$
z_{1} = \sin \alpha + 2i, \quad z_{2} = 1 + \cos \alpha \cdot i \quad (\alpha \in \mathbf{R}).
$$

Then the minimum value of \( f = \frac{13 - \left|z_{1} + i z_{2}\right|^2}{\left|z_{1} - i z_{2}\right|} \) is .

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