
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
Taking a box of matches, I found that I could use them to form any pair of the regular polygons shown in the illustration, using all the matches each time. For example, if I had 11 matches, I could form either a triangle and a pentagon, a pentagon and a hexagon, or a square and a triangle (using only 3 matches for the triangle). However, 11 matches cannot form a triangle with a hexagon, a square with a pentagon, or a square with a hexagon. Of course, each side of the shape must use an equal number of matches.

What is the smallest number of matches that could be in my box?

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