
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
There is one red cell and \( k \) blue cells (\( k > 1 \)), and a deck of \( 2n \) cards numbered from 1 to \( 2n \). Initially, the entire deck is placed in random order in the red cell. From any cell, the top card can be transferred to either an empty cell or on top of a card with a number greater by one. For what maximum \( n \) can the entire deck be transferred to one of the blue cells using these operations?

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