
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 \( a_i, b_i \in \mathbf{C} \) satisfy \( \left|a_i\right|^2 + \left|b_i\right|^2 = 1 \) for \( i = 1, 2, 3 \). Find the minimum value of \( \max_{i \neq j} \left\{ \left| a_i \overline{a_j} + b_i \overline{b_j} \right| \right\} \).

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