
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
In town  $N$  the central square hase a shape of rectangle  $n \times m$ , composed of squares  $1 \times 1$ . In order, to illuminathe the square, lanterns are placed on the corners of the tiles (including the edge of rectangle), such that every lantern illuminates all tiles in corners of which it is placed. Find the minimal amount of lanterns which can be placed, such that every tile will be illuminated even if one of the lanterns burns out.

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