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How about this trip by Make my trip? The trip is only 8 days in total. I have done a trip with them in the past, and they are one of the main tour operators in the country.
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You can try tracking the snow leopard in the Hemis National Park. The trek itself would be for 5-6 days. It would still be difficult to spot a cat. But the trek is usually done in winters since the snow-leopard comes down during that time. Usually, since Ladakh is visited during the summer, you won't have many group departures in winter and it may be difficult to find one.