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I've used Amazon and ebay rebuild kits with success

I must have been looking at the wrong part that you need, this is what I was looking at, look anything like yours. https://www.amazon.com/Wztepeng-31300-HN5-A11-Compatible-2000-2006-430-58002/dp/B09BDVKDSP/ref=sr_1_11?keywords=honda+foreman+es+shift+motor+gear+replacement&qid=1685332521&sr=8-11

I've been cleaning carburetors for the last 60 years. Compressed air and broom straw following a soak in carb cleaner are my goto methods. I have never had gunk so stuck in an orifice that it needed a gas welding tip cleaner or a drill. The tip cleaning kit you show would work fine, but you will need to be careful not to over-do it for risk of enlarging the orifice. I have used drills to enlarge jets, mostly with poor results. Most engines seem to run best with the factory size jets. Hopped up engines are a whole different story. I have been satisfied with factory performance, but I respect the folks who like to hot-rod their equipment. I knew a guy that built sand rails back in the early 70s. He souped up everything, and nobody could come close. I remember he rebuilt a rear end out in the dunes at sand lake on the Oregon coast. He practically had a service station in his trailer. tom