
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  $a, b$  be positive real numbers, and let  $x, y$  be complex numbers such that  $|x| = a$  and  $|y| = b$ . Find the minimal and maximal value of
\[\left|\frac{x + y}{1 + x\overline{y}}\right|\]

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