
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 \[ i^0 \sin 15^\circ + i^2 \cos 45^\circ + i^4 \sin 75^\circ + \dots + i^{40} \cos 345^\circ. \]
A) $\frac{\sqrt{2}}{4}$
B) $\sqrt{2}$
C) $\frac{\sqrt{2}}{2}$
D) $0$
E) $-\frac{\sqrt{2}}{2}$

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