
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 sheet of paper with sides of length $10$ cm is initially folded in half horizontally. The folded paper is then folded diagonally corner to corner, forming a triangular shape. If this shape is then cut along the diagonal fold, what is the ratio of the perimeter of one of the resulting triangles to the perimeter of the original square?
A) $\frac{15 + \sqrt{125}}{40}$
B) $\frac{15 + \sqrt{75}}{40}$
C) $\frac{10 + \sqrt{50}}{40}$
D) $\frac{20 + \sqrt{100}}{40}$

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