
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
Another professor enters the same room and says, "Each of you has to write down an integer between 0 and 200. I will then compute \( X \), the number that is 3 greater than half the average of all the numbers that you will have written down. Each student who writes down the number closest to \( X \) (either above or below \( X \)) will receive a prize." One student, who misunderstood the question, announces to the class that he will write the number 107. If among the other 99 students it is common knowledge that all 99 of them will write down the best response, and there is no further communication between students, what single integer should each of the 99 students write down?

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