
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 hall with a row of 512 closed lockers, numbered from 1 to 512. He starts by opening every second locker. On his return journey, he toggles (opens if closed, closes if open) every third locker. Continuing this pattern, he toggles every fourth locker on his next trip, and repeats this increasing pattern (toggling every \(n^{th}\) locker on the \(n^{th}\) pass). What is the number of the last locker he opens?

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