
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 $|a| = |b| = |c| = 1,$ where $a=1$, $b=-\omega$, $c=-\omega^2$ for $\omega$ being a primitive third root of unity, and
\[az^2 + bz + c = 0.\]
Find the smallest non-zero value of $|z|$.

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