
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 a red card box and $k$ ($k > 1$) blue card boxes, along with a deck of $2n$ cards numbered from 1 to $2n$. Initially, the deck of cards is stacked in the red card box in any order. From any card box, you can take out the top card and either place it in an empty box or place it on top of a card with a number one greater than its own. What is the largest $n$ for which all the cards can be moved to one of the blue card boxes using these operations?

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