
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 the complex number \( z = \cos \theta + \mathrm{i} \sin \theta \) where \( 0^{\circ} \leqslant \theta \leqslant 180^{\circ} \). The complex number \( z \), \( (1+\mathrm{i})z \), and \( 2\bar{z} \) correspond to the points \( P \), \( Q \), and \( R \) on the complex plane, respectively. When \( P \), \( Q \), and \( R \) are not collinear, the fourth vertex of the parallelogram formed with line segments \( PQ \) and \( PR \) as two sides is \( S \). Find the maximum distance from point \( S \) to the origin.

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