
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  $n$  and  $k$  be positive integers satisfying  $k\leq2n^2$ . Lee and Sunny play a game with a  $2n\times2n$  grid paper. First, Lee writes a non-negative real number no greater than  $1$  in each of the cells, so that the sum of all numbers on the paper is  $k$ . Then, Sunny divides the paper into few pieces such that each piece is constructed by several complete and connected cells, and the sum of all numbers on each piece is at most  $1$ . There are no restrictions on the shape of each piece. (Cells are connected if they share a common edge.)
Let  $M$  be the number of pieces. Lee wants to maximize  $M$ , while Sunny wants to minimize  $M$ . Find the value of  $M$  when Lee and Sunny both play optimally. 

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