
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
In a circle, parallel chords of lengths 6, 8, and 10 determine central angles of $\theta$, $\phi$, and $\theta + \phi$ radians, respectively, where $\theta + \phi < \pi$. Determine $\cos \theta$, which is also a positive rational number expressed as a fraction in lowest terms. What is the sum of its numerator and denominator?

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