
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
After lunch, there were dark spots on a transparent square tablecloth with a total area $S$. It turned out that if the tablecloth is folded in half along any of the two lines connecting the midpoints of its opposite sides, or along one of its two diagonals, then the total visible area of the spots will be equal to $S_{1}$. However, if the tablecloth is folded in half along the other diagonal, the total visible area of the spots remains equal to $S$. What is the smallest value that the ratio $S_{1}: S$ can take?

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