
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 $a$, $b$, $c$, $z$ be complex numbers such that $|a| = |d| > 0$, $b = kd$, $c = k^2d$ for some real number $k$, and
$$az^2 + bz + c = 0.$$
Find the largest possible value of $|z|.$

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