
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 student walks down a corridor that houses a sequence of closed lockers, numbered from 1 to 512. He starts by toggling every second locker (starting with locker 2, so he opens locker 2, 4, 6, ...). When he reaches the end, the student turns around and toggles every third locker (starting with the first locker in the sequence that remains closed). He continues this pattern, increasing the step by one each time he reaches the end of the corridor and turns around, until all lockers are open. What is the number of the last locker he opens?

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