
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 \(\operatorname{Re}(z_{1}) > 0\) and \(\operatorname{Re}(z_{2}) > 0\) 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}
user