
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
Consider the sum \[\cos{60^\circ} + i\cos{150^\circ} + \cdots + i^n\cos{(60 + 90n)^\circ}  + \cdots + i^{30}\cos{2760^\circ}\]. Given $i^2 = -1$, compute the simplified expression for this sum.
A) \( 8 - 7i \)
B) \( 7.5 - 7i \)
C) \( 7.5 - 8i \)
D) \( 8.5 - 7i \)

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