
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
Suppose points $P$, $Q$, and $R$ are represented in the complex plane by complex numbers $z$, $(2 + 2i)z$, and $3 \overline{z}$, respectively, where $|z| = 1$. When $P$, $Q$, and $R$ are not collinear, find the maximum distance from the origin to the fourth vertex $S$ of the parallelogram $PQSR$.

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