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Mehadrin vs. Non Mehadrin

I am preparing a chabura on the leniencies that the Rabbanut in Israel takes. With the understanding that the various Rabbanut branches around the country have different standards, this can include general leniencies that they all take, or that individual branches (e.g. the Rabbanut of Beit Shemesh) take. Also, I am interested in the specific differences between standard Rabbanut and Rabbanut Mehadrin. Referring me to sources that the leniencies are based on would also be greatly appreciated.