Yogurt Cake with Currant Raspberry Sauce

Name: Yogurt Cake with Currant Raspberry Sauce
Ingredients: 3 cups sifted cake flour (not self-rising; sift before measuring); 2 teaspoons baking powder; 1 teaspoons baking soda; 1/2 teaspoon salt; 2 sticks unsalted butter, softened; 2 cups granulated sugar; 2 teaspoons pure vanilla extract; 3 large eggs, warmed in shell in warm water 10 minutes; 2 cups well-stirred plain whole-milk yogurt (not Greek-style) atroom temperature 30 minutes; 2 cups fresh red currants (1/2 pound), stemmed; 4 cups fresh raspberries (16 ounces), divided; 2/3 cup granulated sugar room temperature 30 minutes; 3 cups confectioners sugar; 3 tablespoons light corn syrup; 1/2 cup heavy cream; Scant 1/2 teaspoon pure vanilla extract; Equipment: a 13- by 9-inch cake pan