
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 $P$, $Q$, and $R$ be points in the complex plane represented by the complex numbers $z$, $(2+i)z$, and $2\overline{z}$, respectively, where $|z|=1$. If $P$, $Q$, and $R$ are not collinear, and $S$ corresponds to the fourth vertex of the parallelogram $PQRS$, determine the maximum distance between $S$ and the origin.

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