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Preheat oven to 375 degrees F.
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Spray cookie sheets with nonstick cooking spray.
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In medium bowl, sift together flour, ginger, baking soda, allspice, and salt.
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In large bowl, cream together sugar and margarine until well-blended.
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Stir in applesauce and molasses.
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Add flour mixture to applesauce mixture; mix until well blended.
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Drop dough by rounded tablespoons 1 inch apart onto prepared cookie sheets.
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Flatten each slightly with moistened fingertips.
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Bake 11-14 minutes, or until firm.
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Watch carefully at end of baking, to prevent overbrowning.