
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 \( z_{n} = \left(\frac{1-\mathrm{i}}{2}\right)^{n}, n \in \mathbf{N}_{+} \). Define \( S_{n} = \sum_{k=1}^{n}\left|z_{k+1} - z_{k}\right| \). Find \( \lim_{n \rightarrow \infty} S_{n} \).

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