
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 \( z_{1} \) and \( z_{2} \) satisfy \( \operatorname{Re} z_{1} > 0 \), \( \operatorname{Re} z_{2} > 0 \), and \( \operatorname{Re} z_{1}^{2} = \operatorname{Re} z_{2}^{2} = 2 \). Find:
(1) The minimum value of \( \operatorname{Re} (z_{1} z_{2}) \);
(2) The minimum value of \( \left| z_{1} + 2 \right| + \left| \overline{z_{2}} + 2 \right| - \left| \overline{z_{1}} - z_{2} \right|.

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