
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 \( z_{1} = -3 - \sqrt{3} \, \mathrm{i} \), \( z_{2} = \sqrt{3} + \mathrm{i} \), and \( z = \sqrt{3} \sin \theta + \mathrm{i} (\sqrt{3} \cos \theta + 2) \). Find the minimum value of \( \left| z - z_{1} \right| + \left| z - z_{2} \right| \).

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