
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
The points \( P \), \( Q \), and \( R \), on the complex plane, are represented by the complex numbers \( z \), \( (2+i)z \), and \( 3\overline{z} \), respectively. Assuming \( |z| = 1 \), and when \( P \), \( Q \), and \( R \) are not collinear, determine the maximum distance between the fourth vertex \( S \) of the parallelogram \( PQRS \) and the origin.

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