
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
Let \( z = e^{2 \pi i / n} = \cos \left(\frac{2 \pi}{n}\right) + i \sin \left(\frac{2 \pi}{n}\right) \). For an arbitrary integer \( a \), calculate the sums

a) \( 1 + z^a + z^{2a} + \ldots + z^{(n-1)a} \)

b) \( 1 + 2z^a + 3z^{2a} + \ldots + nz^{(n-1)a} \).

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