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Pour yeast into a small bowl; add warm water and 1/4 cup of flour. Stir, cover with two damp kitchen towels, and leave in a warm place until bubbly, about 2 1/2 hours. Combine butter, shortening, and mastic in a saucepan over medium-low heat. Stir milk and sugar together in another saucepan over medium-low heat until sugar is dissolved and mixture is warm. Remove from heat. Pour 2 cups flour into the butter mixture and stir well. Add salt, then stir in the yeast mixture. Add 2 more cups of flour and stir well. Stir in 3 beaten eggs. Slowly add the milk mixture and 1 cup of flour and stir well. Add the room-temperature water and the last cup of flour and stir until the dough is not sticky. Knead dough on a lightly floured surface for 20 minutes with lightly floured hands. Form dough into a ball. Cover with damp towels and let rise in a warm place until doubled in volume, about 2 hours. Preheat the oven to 325 degrees F (165 degrees C). Grease two baking sheets. Divide dough into two pieces and separate each piece into three strands. Braid each of the 3 strands together to form 2 braided loaves. Place on the baking sheets. Nestle the red-dyed eggs between the strands. Brush the dough with the remaining beaten egg. Sprinkle the tops with almonds and sesame seeds. Bake in the preheated oven until golden brown, about 25 minutes.