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Kitchen sink blockage

Kitchen sink isn't draining at all, although all other plumbing fixtures (adjacent dishwasher for example seem to drain just fine). I've cleaned out the U-bend, and also hand-snaked it for 4m and it seems clear, but still backs up when I connect up again! Either it needs a much longer snake, or I've overlooked something. Either way, needs someone who knows what they're doing. We're on a septic tank, which seems to be just fine.