
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 containing a row of closed lockers, numbered $1$ to $2048$. He opens every third locker starting from locker number 1. After reaching the end, he turns around and opens the first closed locker he encounters, then skips two lockers, and opens the next closed one, continuing this pattern. On subsequent trips, he alters the pattern: if a locker is closed, he opens it after skipping as many lockers as the last number he opened mod 16. When no lockers remain closed, which is the last locker he opens?

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