
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
Derive the formula $\cos ((n+1) \alpha) = 2 \cos \alpha \cos (n \alpha) - \cos ((n-1) \alpha)$, where $n$ is any real number, and use it to express $\cos (3 \alpha)$ and $\cos (4 \alpha)$ as polynomials in terms of $\cos \alpha$.

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