
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
There are \(N \geq 9\) different non-negative numbers less than one written on the board. It turned out that for any eight different numbers from the board, there exists a ninth number, different from them, such that the sum of these nine numbers is an integer. For which \(N\) is this possible?

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