
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
Calculate the sum of the series \[i\cos{30^\circ} + i^2\cos{120^\circ} + \cdots + i^n\cos{(30 + 90n)^\circ} + \cdots + i^{30}\cos{2730^\circ}.\]
A) \(-\frac{13}{2} + \frac{\sqrt{3}}{2} - \frac{i}{2}\)
B) \(-7\sqrt{3} + i\)
C) \(-14 - 7i\sqrt{3}\)
D) \(\frac{\sqrt{3}}{2}(7 - 3i)\)

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