Today, when the correct time was $5$:$00$am$,$ my analog clock was showing $4$:$57($my clock does not give any indication of AM or PM$).$ The clock also runs a little slow, and loses exactly $7$ minutes every hour. So$,$ an hour later, when the correct time is $6$:$00,$ my clock shows $5$:$50,$ and two hours later, when the correct time is $7$:$00,$ my clock shows $6$:$43.$

How many days after today is the earliest instance when my analog clock shows the correct time on the hour. i$.$e$.$ when the correct time is $x$:$00($or $xY$:$00),$ and my analog clock also shows the same time.

Don't forget: the clock does not show AM$/$PM$.$

You need to provide your answer as just a single whole number, indicating the number of days after today when the clock shows correct time on the hour. To solve this question, you can use any tool, or website, or google, or any software package, or any programming language, or tool. However, you're not allowed to discuss this with any other person, or ask for advice.