
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]


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Let \( n \) and \( k \) be fixed positive integers of the same parity, with \( k \geq n \). We are given \( 2n \) lamps numbered 1 through \( 2n \); each of them can be on or off. At the beginning, all lamps are off. We consider sequences of \( k \) steps where, at each step, one of the lamps is switched (either from off to on or from on to off).

Let \( N \) be the number of \( k \)-step sequences that end with lamps \( 1 \) through \( n \) on, and lamps \( n+1 \) through \( 2n \) off.

Let \( M \) be the number of \( k \)-step sequences that lead to the same final state and do not touch lamps \( n+1 \) through \( 2n \) at all. 

Find the ratio \( \frac{N}{M} \).

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