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Pushing the cork down into the bottle is the easiest way to get to your drink, but it's also the messiest to deal with. If your cork has broken and you can't pull it out using other methods, you can always push it in. Before you go pushing the cork into the bottle, make sure to remove any debris that you can from the cork. You will always end up with some, but try to make it easier on yourself and get rid of as much as possible. Make sure to do this somewhere you are ok with possibly wine or champagne squirting out of the bottle. You should not be wearing any clothing you like for this method. The pressure that you release when pushing the cork down into the bottle may cause some of the liquid to spray out. Using your fingers, push the cork down until it falls into the bottle. You will now have access to your drink, but there is also a cork and some debris in the wine or champagne. Quickly strain the liquid. After the cork lands in the bottle, use a coffee filter to strain out the residual cork pieces. If you have a glass coffee pot like a Chemex with paper filters you can pour your wine from your bottle into the container. The filter will catch all the cork debris and let the liquid pass through. You can also use any paper filter over any type of container. You can rinse the bottle and pour the wine back in using a funnel. You may want to use another container, though. If most of the cork is still in the bottle, you can also just pour the wine into a decanter. Then, enjoy. Once you push a cork into a bottle, you won't be able to easily remove it. It's best to store your liquid in a new bottle.
Remove any debris from the cork. Push the cork into the bottle. Pour your wine through a coffee filter or strainer. Pour the wine into a new container.