
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 $p, q, r$ be distinct positive integers, and let $\zeta$ be a complex number such that $\zeta^4 = 1$ and $\zeta \neq 1.$ Find the smallest possible value of
\[
|p + q \zeta + r \zeta^3|.
\]

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