
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
Using a sequence to traverse all integral points on the complex plane: Let \( a_{0}=0 \), \( a_{1}=1 \), and then move one square at a time in a counterclockwise direction. Further, let \( a_{n+1}-a_{n} = \mathrm{i}^{f(n)} \), where \( \mathrm{i} \) is the imaginary unit. Find the simplest unified expression for \( f(n) \).

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