
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
Find the minimal natural number \( n \) with the following property: It is possible to tile the plane with squares whose side lengths belong to the set \(\{1, 2, \ldots, n\}\) so that no two squares with the same side length touch along a segment of an edge.

*Remark:* Squares with the same side length can touch at a vertex. Not all of the side lengths from 1 to \( n \) need to be used in the tiling.

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