
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
Find the sums of the following infinite series:

a) \(1 + a + a^2 + \ldots + a^n + \ldots\)

b) \(a + 2a^2 + 3a^3 + \ldots + na^n + \ldots\)

c) \(\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \ldots + \frac{1}{n(n+1)} + \ldots\)

d) \(\frac{1}{1 \cdot 2 \cdot 3} + \frac{1}{2 \cdot 3 \cdot 4} + \frac{1}{3 \cdot 4 \cdot 5} + \ldots + \frac{1}{n(n+1)(n+2)} + \ldots\)

e) \(1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} + \ldots\)

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