
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 $a,$ $b,$ $c,$ $d,$ $z,$ and $w$ be complex numbers such that $|a| = |b| = |c| = |d| > 0$ and
\[ az^3 + bwz^2 + cz + d = 0. \]
Find the largest possible value of $|z|$ given $|w| = \frac{1}{2}$.

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