
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
Evaluate the sum \[\cos{30^\circ} + i\cos{60^\circ} + \cdots + i^n\cos{(30 + 30n)^\circ}  + \cdots + i^{45}\cos{1380^\circ}\].
A) $\frac{11\sqrt{3}}{2} + \frac{33i}{2}$\
B) $\frac{11\sqrt{3}}{2} - \frac{33i}{2}$\
C) $11\sqrt{3} + 33i$\
D) $11\sqrt{3} - 33i$

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