
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 sums:

a) \(1 + a \cos \varphi + \ldots + a^{k} \cos k \varphi + \ldots \) \(\left(|a| < 1\right)\)

b) \(a \sin \varphi + \ldots + a^{k} \sin k \varphi + \ldots \) \(\left(|a| < 1\right)\)

c) \(\cos \varphi + C_{n}^{1} \cos 2 \varphi + \ldots + C_{n}^{n} \cos (n+1) \varphi\)

d) \(\sin \varphi + C_{n}^{1} \sin 2 \varphi + \ldots + C_{n}^{n} \sin (n+1) \varphi\)

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