
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
 $N\geq9$  distinct real numbers are written on a blackboard. All these numbers are nonnegative, and all are less than  $1$ . It happens that for very  $8$  distinct numbers on the board, the board contains the ninth number distinct from eight such that the sum of all these nine numbers is integer. Find all values  $N$  for which this is possible. *(F. Nilov)*

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