
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
A square is cut into red and blue rectangles. The sum of the areas of the red rectangles is equal to the sum of the areas of the blue rectangles. For each blue rectangle, write down the ratio of the length of its vertical side to the length of its horizontal side, and for each red rectangle, write down the ratio of the length of its horizontal side to the length of its vertical side. Find the smallest possible value of the sum of all recorded numbers.

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