
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
Suppose there is an infinite sequence of lights numbered \(1,2,3, \ldots\), and you know the following two rules about how the lights work:
- If the light numbered \(k\) is on, the lights numbered \(2k\) and \(2k+1\) are also guaranteed to be on.
- If the light numbered \(k\) is off, then the lights numbered \(4k+1\) and \(4k+3\) are also guaranteed to be off.
Suppose you notice that light number 2023 is on. Identify all the lights that are guaranteed to be on?

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