
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

Simplify

$$
\begin{aligned}
\cos \alpha + C_{n}^{1} \cos 2 \alpha + C_{n}^{2} \cos 3 \alpha + \ldots + C_{n}^{n-1} \cos n \alpha + \cos (n+1) \alpha
\end{aligned}
$$

and

$$
\sin \alpha + C_{n}^{1} \sin 2 \alpha + C_{n}^{2} \sin 3 \alpha + \ldots + C_{n}^{n-1} \sin n \alpha + \sin (n+1) \alpha
$$

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