
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
Let \( n \) and \( k \) be positive integers such that \( k \geq n \) and \( k - n \) is an even number. We are given \( 2n \) lamps numbered from 1 to \( 2n \), each of which can be either on or off. Initially, all the lamps are off. We are considering a sequence of steps, where each step consists of changing the state of one lamp (from on to off or from off to on).

Let \( N \) be the number of sequences of \( k \) steps that result in the lamps numbered from 1 to \( n \) being on and the lamps numbered from \( n+1 \) to \( 2n \) being off.

Let \( M \) be the number of sequences of \( k \) steps that result in the lamps numbered from 1 to \( n \) being on and the lamps numbered from \( n+1 \) to \( 2n \) being off, and no lamps numbered from \( n+1 \) to \( 2n \) are ever turned on at any point in the sequence.

Determine the value of the ratio \( N / M \).

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