
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
A finite collection of positive real numbers (not necessarily distinct) is *balanced*if each number is less than the sum of the others. Find all  $m \ge 3$  such that every balanced finite collection of  $m$  numbers can be split into three parts with the property that the sum of the numbers in each part is less than the sum of the numbers in the two other parts.

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