
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  $M, \alpha, \beta \in \mathbb{R} $  with  $M > 0$  and  $\alpha, \beta \in (0,1)$ . If  $R>1$  is a real number, we say that a sequence of positive real numbers  $\{ C_n \}_{n\geq 0}$  is  $R$ -*inoceronte* if  $ \sum_{i=1}^n R^{n-i}C_i \leq R^n \cdot M$  for all  $n \geq 1$ . Determine the smallest real  $R>1$  for which exists a  $R$ -*inoceronte* sequence  $ \{ C_n \}_{n\geq 0}$  such that  $\sum_{n=1}^{\infty} \beta ^n C_n^{\alpha}$  diverges.

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