
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
Given a  $n\times m$  grid we play the following game . Initially we place  $M$  tokens in each of  $M$  empty cells and at the end of the game we need to fill the whole grid with tokens.For that purpose  we are allowed to make the following move:If an empty cell shares a common side with at least two other cells that contain a token then we can place a token in this cell.Find the minimum value of  $M$  in terms of  $m,n$  that enables us to win the game.

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