
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 all positive integer  $n(\ge 2)$  and rational  $\beta \in (0,1)$  satisfying the following:
There exist positive integers  $a_1,a_2,...,a_n$ , such that for any set  $I \subseteq \{1,2,...,n\}$  which contains at least two elements,  $$  S(\sum_{i\in I}a_i)=\beta \sum_{i\in I}S(a_i).  $$  where  $S(n)$  denotes sum of digits of decimal representation of  $n$ .

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