
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
At the round table there are  $10$  students. Every of the students thinks of a number and says that number to its immediate neighbors (left and right) such that others do not hear him. So every student knows three numbers. After that every student publicly says arithmetic mean of two numbers he found out from his neghbors. If those arithmetic means were  $1$ ,  $2$ ,  $3$ ,  $4$ ,  $5$ ,  $6$ ,  $7$ ,  $8$ ,  $9$  and  $10$ , respectively, which number thought student who told publicly number  $6$ 

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