
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
\[ 3.408 \cos ^{2} \alpha + \cos ^{2} 2 \alpha + \ldots + \cos ^{2} n \alpha = \frac{\cos (n+1) \alpha \cdot \sin n \alpha}{2 \sin \alpha} + \frac{n}{2}. \]

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