
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 complex numbers \( z_1 \) and \( z_2 \) satisfy \(\operatorname{Re}(z_1) > 0\), \(\operatorname{Re}(z_2) > 0\), and \(\operatorname{Re}(z_1^2) = \operatorname{Re}(z_2^2) = 2\) (where \(\operatorname{Re}(z)\) denotes the real part of the complex number \(z\)).

(1) Find the minimum value of \(\operatorname{Re}(z_1 z_2)\).

(2) Find 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}
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