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If the cut is actively bleeding, the first step is to stop that bleeding.  Do this by applying pressure to the area, using a clean cloth or medical gauze.  Do not remove the cloth until the bleeding has stopped completely.  Facial injuries often bleed worse than other parts of the body, so the injury may look worse than it is.  Crying makes bleeding worse, so try to remain calm and stop crying. If the cut is very deep, particularly if it is a puncture wound, you may need to go to the hospital immediately.  Large gaping wounds or deep wounds will likely need stitches and professional cleaning.  More superficial wounds can be cared for at home. Before you touch the open wound in any way, be sure to clean your hands thoroughly with soap and hot water.  Wash both hands, between all of your fingers, and your wrists thoroughly, rinse with hot water, and dry with a clean towel. Hand-washing is a very important method of avoiding the chance of any infection in the wound. Clean the wound very gently with water and soap.  Be sure to rinse the soap out of the wound completely with water.  Be sure to remove any visible debris or dirt from the wounded area.  Use cool or slightly warm water.  Water that is too hot may cause the wound to start bleeding again.   Be patient and slow with this step.  If there is debris in the wound, try using a soft cloth to aid in removing it. If necessary, sterilize tweezers with rubbing alcohol and use them to aid in removing pieces of debris from the wound.   Avoid using hydrogen peroxide or iodine, which can irritate or damage tissue. An antibiotic ointment such as Neosporin or Polysporin is the best option, but if none is available, simple petroleum jelly such as Vaseline can help.  Expensive creams or treatments that claim to reduce scarring are generally not as helpful as they claim to be. Place a sterile bandage over the wounded area.  This may be a bit cumbersome on your face, but it is important to keep the area free from potential infections.  Place a bandage over the cut and use medical tape above and below the bandage to keep it in place. If the wound is still bleeding, try to secure the bandage tightly over the area.  If it is not, a loose covering is adequate. A wide open cut needs to be pinched together to aid in healing and reduce scarring.  Butterfly tape can help pull the skin together and enable it to heal.  If this does not work, you will likely need stitches and you should go to the hospital. If the wound area is swollen (for example, if the cut is the result of a forceful blow), it is important to get the swelling in the area down as well.  This can be done by putting ice on the area for 20 minutes at a time.
Stop any bleeding. Assess the injury. Wash your hands. Wash out the cut completely. Apply treatment to the wound. Bandage the wound. Use butterfly tape for wide wounds. Reduce any swelling.