How is Yom Kippur like Purim?

A midrash compares the two days, by noting that the most solemn Fast Day on the Jewish calendar, Yom Kippur, is similar to Purim: the day of feasting, laughter and sending packages of goodies to  friends and to the poor. All via one  letter: the Hebrew letter kaf, which in this case, is used in the name of the Fast Day: "k"  setting up a comparison of  "almost" opposites. Purim was a day that could have gone either way, like Yom Kippur: we as a people could have been wiped out by those who had convinced a mortal king that we were his enemies on Purim, and on Yom Kippur, our fate likewise hangs in the balance, but as individuals pleading to God regarding their own destiny. Purim begins with a fast the day before, comemmorating Esther's fast, and ends with a victorious feast, referenced above. Yom Kippur is an evening to evening fast, comprising 25 hours to make sure that  everyone who wants to ask for forgiveness, first from the person wronged, and then from God, will have enough time to do so. In fact, they will have had the Aseret Y'mei Teshuvah in which to do this plus the books are not really closed at the end of the fast, just in case we suddenly remember something or someone about which or from whom we needed to seek forgiveness. We break the fast in community, and then we are supposed to use the extra spiritual (and caloric) rush to do a mitzvah, a commandment. Theory is fine, but putting it into practice is the whole point!

NOTE: I will be out of the country for the High Holy Days in 2024 and unable to make any kippot during that time. Feel free to browse and ask questions:  I'll check in during the week between Rosh Hashanah and Yom Kippur.