
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
Let \(ABC\) be a triangle with \(AB = 13\), \(BC = 14\), and \(CA = 15\). Let \(I_A\), \(I_B\), and \(I_C\) be the \(A\), \(B\), and \(C\) excenters of this triangle, respectively, and let \(O\) be the circumcenter of the triangle. Let \(\gamma_A\), \(\gamma_B\), and \(\gamma_C\) be the corresponding excircles and \(\omega\) be the circumcircle. \(X\) is one of the intersections between \(\gamma_A\) and \(\omega\). Likewise, \(Y\) is an intersection of \(\gamma_B\) and \(\omega\), and \(Z\) is an intersection of \(\gamma_C\) and \(\omega\). Compute
\[
\cos \angle OXI_A + \cos \angle OYI_B + \cos \angle OZI_C.
\]

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