
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
Given the imaginary unit \( \mathrm{i} \), determine the natural number \( m \) that satisfies the equation \( \sum_{k=1}^{m} \mathrm{i}^{k}=\prod_{k=1}^{m} \mathrm{i}^{k} \). Construct a sequence \( \left\{a_n\right\} \) with these numbers arranged in ascending order. What is the general term of the sequence \( \left\{a_n\right\} \)?

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