
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
Given that \( i^{2} = -1 \), express the sum \( S=\cos 45^{\circ}+i \cos 135^{\circ}+\cdots+i^{n} \cos (45+90n)^{\circ}+\cdots+i^{40} \cos 3645^{\circ} \) in simplest form.

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