
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
Consider the curve $y = \cos x$ and the line $y = \cos 30^\circ$. Determine the ordered pair of relatively prime positive integers $(p, q)$ that represents the ratio of lengths of segments when the curve cuts the line, maintaining successive ratios: $\dots p : q : p : q \dots$ with $p < q$.

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