Sending attachment to Chat customers

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I know that attachments for chat are only supported incoming.  So currently if we are engaged in a live chat and the agent needs to send an attachment to the customer, they create a new incident for that customer and send the attachment via this incident. This is an additional incident to the actual chat incident that get created, since the chat is not finished yet at this point.  This seems like such a cumbersome process and I was wondering if there is a better way of doing this.  Any ideas or best practices?