
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, four parallel chords of lengths 3, 4, 5, and 12 align such that their subtended central angles are $\phi$, $\psi$, $\theta$, and $\phi + \psi + \theta$, respectively, where $\phi + \psi + \theta < \pi$. If $\cos \phi$ is a positive rational number expressed as a fraction in simplest terms, what is the sum of its numerator and denominator?

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