
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, dark spots with a total area of \( S \) remained on a transparent square tablecloth. It was found that if you fold the tablecloth in half along any of the two lines connecting the midpoints of opposite sides, or along one of the two diagonals, the total visible area of the spots would be \( S_1 \). However, if you fold the tablecloth in half along the other diagonal, the total visible area of the spots remains \( S \). What is the smallest possible value of the ratio \( \frac{S_{1}}{S} \)?

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