
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
Let  $n$  a positive integer. In a  $2n\times 2n$  board,   $1\times n$  and  $n\times 1$  pieces are arranged without overlap. 
Call an arrangement **maximal** if it is impossible to put a new piece in the board without overlapping the previous ones.
Find the least  $k$  such that there is a **maximal** arrangement that uses  $k$  pieces.

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