
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
Find the ratio of line segments for the curve $y = \sin x$ when it cuts the line $y = \sin 60^\circ$. Assume the curve intersects this line in a pattern yielding segments with ratios \[\dots p : q : p : q \dots\] where $p < q$. Compute the ordered pair $(p,q)$ of relatively prime positive integers representing these ratios.

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