
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 the complex numbers \( a \), \( b \), and \( c \) satisfy the condition: for any complex number \( \mathrm{z} \) with a modulus not exceeding 1, the inequality \( \left|\mathrm{az}^{2}+\mathrm{bz}+\mathrm{c}\right| \leq 1 \) holds. Find the maximum value of \( |b c| \).

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