
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  $\alpha>0$ , consider the infinite real sequence defined by  $x_1=1$  and 
\[ \alpha x_n = x_1+x_2+\cdots+x_{n+1}  \mbox{\qquad for } n\ge1. \] Determine the smallest  $\alpha$  for which all terms of this sequence are positive reals.
(Proposed by Gerhard Woeginger, Austria)

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