
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
It is given positive integer  $n$ . Let  $a_1, a_2,..., a_n$  be positive integers with sum  $2S$ ,  $S \in \mathbb{N}$ . Positive integer  $k$  is called separator if you can pick  $k$  different indices  $i_1, i_2,...,i_k$  from set  $\{1,2,...,n\}$  such that  $a_{i_1}+a_{i_2}+...+a_{i_k}=S$ . Find, in terms of  $n$ ,  maximum number of separators

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