
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:
\[
i\sin{30^\circ} + i^2\sin{120^\circ} + i^3\sin{210^\circ} + \ldots + i^{30}\sin{(90n-60)^\circ}
\]
where \( n \) is the nth term in the summation from 1 to 30.
A) \(7.5 - 3.75\sqrt{3}\)   
B) \(7.5 - 3.75\sqrt{3} - 3.75i\sqrt{3}\)    
C) \(3.75\sqrt{3} - 3.75i\sqrt{3}\)    
D) \(3.75i\sqrt{3} - 3.75\sqrt{3}\)

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