
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
The curve $y = \sin x$ intersects the horizontal line $y = \sin 50^\circ$. If these intersections occur at successive $x$ values alternating in a specific pattern, what are the ratios of the lengths of these segments? Assume the ratio alternates as $r:s:r:s\dots$ with $r < s$. Compute the ordered pair of relatively prime positive integers $(r, s)$.

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