
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 with lockers numbered from $1$ to $512$. Initially, all lockers are closed. The student opens the first locker, then skips one and opens the next, and so on until the end of the corridor. On reaching the end, the student turns around, opens the first closed locker, skips one closed locker, opens the next, and continues in this manner. The student repeats this process until all lockers are open. Identify the number of the last locker to be opened.

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