
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
Define a kind of operation for complex numbers \\(z_{1} * z_{2}= \frac{|z_{1}|+|z_{2}|}{2}\\) (the right side of the equation is a normal operation), if the complex number \\(z=a+bi\\), and the positive real numbers \\(a\\), \\(b\\) satisfy \\(a+b=3\\), then find the minimum value of \\(z*\overline{z}\\).

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