
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 are one red cell and \( k \) (with \( k > 1 \)) blue cells, as well as a deck of \( 2n \) cards numbered from 1 to \( 2n \). Initially, the entire deck is in random order in the red cell. From any cell, you can take the top card and move it either to an empty cell or onto a card whose number is greater by one. What is the maximum \( n \) for which it is possible to move the entire deck into one of the blue cells using these operations?

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