Rinse the chicken breasts and place them in a large stockpot with 12 cups of cold water. Add the garlic cloves, half of the chopped onion, bay leaves, salt, and black pepper. Bring to a boil over high heat, then reduce to medium-low and simmer for 45 minutes, skimming off any foam that rises to the surface.

While the chicken cooks, peel all three varieties of potatoes. Cut the criolla potatoes into thin rounds (they will dissolve to thicken the soup). Cut the pastusa potatoes into 1-inch chunks and the sabanera potatoes into quarters. Keep each variety separate.

Remove the chicken from the pot and set aside to cool. Strain the broth through a fine-mesh sieve, discarding the solids. Return the clear broth to the pot.

Add the criolla potato slices to the broth and bring to a boil. Cook for 15-20 minutes, stirring occasionally, until the potatoes completely dissolve and thicken the soup.

Add the pastusa and sabanera potato chunks, corn pieces, remaining chopped onion, and guascas (or cilantro and oregano substitute). Season with additional salt to taste. Simmer for 25-30 minutes until all potatoes are tender but not falling apart.

While the soup finishes cooking, shred the cooled chicken into bite-sized pieces, discarding the skin and bones.

Return the shredded chicken to the pot and simmer for 5 more minutes to heat through. Taste and adjust seasoning.

Ladle the hot ajiaco into deep bowls, ensuring each serving has chicken, all types of potatoes, and corn. Serve immediately with small bowls of capers, heavy cream, avocado slices, and white rice on the side for guests to customize their bowls.