
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
We are given one red and  $k>1$  blue cells, and a pack of  $2n$  cards, enumerated by the numbers from  $1$  to  $2n$ .  Initially, the pack is situated on the red cell and arranged in an arbitrary order.  In each move, we are allowed to take the top card from one of the cells and place it either onto the top of another cell on which the number on the top card is greater by  $1$ , or onto an empty cell.  Given  $k$ , what is the maximal  $n$  for which it is always possible to move all the cards onto a blue cell?

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