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"Whether you want to walk for one mile, 5 miles or 20 miles, the Queen Elizabeth Park, just off the A3 outside Petersfield, is a good choice for all walkers. It's a huge and interesting park...20 miles of trails for walkers, cyclists & horse riders...and apparently it's the highest point on the south downs.
I really enjoyed the 5 mile walk (took about 1 and 3/4 hours), but you can take the 1 mile route or get maps in the shops for longer treks. Aside from the scenery and fresh air, the best thing about this park is that it is clearly laid out with instructions all the way round so you can't get lost!
Also great for horseriders and cyclists as there are specials tracks for them.
How to find it: The Queen Elizabeth Country Park is in South East Hampshire 4 miles south of Petersfield
The site has its own sliproads on both the north and south bound carriageways of the A3."