
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 \( z \in \mathbf{C} \), and let \( f(z)=a z^{2}+b z+c \) be a polynomial with complex coefficients. Given that \( |f(z)| \leq 1 \) when \( |z| \leq 1 \), what is the range of \( |a z + b| \) when \( |z| \leq 1 \)?

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