
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
We call the sum of any  $k$  of  $n$  given numbers (with distinct indices) a  $k$ -sum. Given  $n$ , find all  $k$  such that, whenever more than half of  $k$ -sums of numbers  $a_{1},a_{2},...,a_{n}$  are positive, the sum  $a_{1}+a_{2}+...+a_{n}$  is positive as well.

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