
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 partial sum of the first $n$ terms $\left(S_{n}\right)$ for each series; demonstrate the convergence (or divergence) of the series using the definition; find the sum of the series $S$:
1) \( a + aq + aq^2 + \ldots + aq^{n-1} + \ldots \)

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

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