
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
For a real number $\alpha > 1$, define an infinite sequence $\left\{a_{n}\right\}_{n \in \mathbb{N}}$ such that $a_{n}=\left\lfloor \alpha^{n} \right\rfloor$. If for any positive integer $m$, there exist a positive integer $k$ and $k$ natural numbers $0 \leq i_{1} < i_{2} < \ldots < i_{k}$ such that $m = \sum_{j=1}^{k} a_{i_{j}}$, find all possible values of $\alpha$.

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