
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 bored student walks down a hall that contains a row of closed lockers, numbered 1 to 1024.  He opens the locker numbered 1, and then alternates between skipping and opening each closed locker thereafter.  When he reaches the end of the hall, the student turns around and starts back.  He opens the first closed locker he encounters, and then alternates between skipping and opening each closed locker thereafter. The student continues wandering back and forth in this manner until every locker is open.  What is the number of the last locker he opens?

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