
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 rectangle is divided by lines parallel to its sides into squares with side length 1, which are colored in a chessboard pattern with white and black squares. The diagonal of the rectangle is split into white and black segments. Find the ratio of the sum of the lengths of the white segments to the sum of the lengths of the black segments if the dimensions of the rectangle are: a) $100 \times 99$; b) $101 \times 99$.

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