
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 school has 100 students and 100 lockers numbered from 1 to 100. Initially, all the lockers are closed. The first student passes and opens all the lockers; the second passes and closes all the even-numbered lockers; the third passes and toggles the position of all lockers that are multiples of 3, meaning those that are open become closed and those that are closed become open; the fourth student toggles the position of all lockers that are multiples of 4; and so on until the one hundredth student, who toggles the position of the lockers that are multiples of 100. After all the students have passed, how many lockers remain closed?

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