
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 \(\mathcal{F}\) be the family of all nonempty finite subsets of \(\mathbb{N} \cup \{0\}\). Find all positive real numbers \(a\) for which the series

\[ 
\sum_{A \in \mathcal{F}} \frac{1}{\sum_{k \in A} a^{k}} 
\]

is convergent.

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