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I've used a variety of razors in my days, including the Mach 3. Primarily due to the fact that I have a very hard beard and can only shave against the grain, in order to have a close enough shave.
At this time I'm using a Wilkinson Sword Quattro, closest shave I've ever gotten and never cut myself. Try it, not exactly expensive. Oh and you may also look into some shaving oil to apply before applying foam/gel. It increases the glide significantly and reduces the risk of cutting yourself. Plus it softens the beard so it's easier to shave it off.
Also, if you have problems with razor burn and irritated skin after shaving, get an after shave balm. It reduces razor burn and if your beard is not as hard as mine chances are it will eliminate it completely.
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