[question] Promoting with K+N+P vs K+N

I'm playing throught the computer workout and am practicing these endgames. I reach a point where the computer tells me that I have "mate in 20." All moves except for Na5 at this point add to the game counter (or draw). I sort of understand it, but can someone please explain to me what is going on here? What is the goal I am supposed to keep in the back of my mind so that I don't get an accidental fork? Is there a pattern or a sequence that is best memorised?

Edit: I just noticed that 10.Na7 gains the queen 5 moves earlier than the "best line." Shouldn't this logically ead to a quicker mate?

So does nobody know... ???

Clearly black's main resource for when his knight is removed from f7 is to play it to g5, threatening to win the new queen with a fork, and white needs to have something planned against this.

The Na5 line is more flexible than the Na7 line because it can go to both d8 and d6. In the Na7 line there is only the option of going to d6. 

However this seems like a poor training tool that you are using because it only shows the theoretically "best" moves, i.e. those which delay mate the longest, not those which pose the most problems for the side trying to win. In the 10.Na7 line 12...Ng5 is much more testing, because even though it theoretically loses quicker, white still has to find a way to promote the pawn and it's not as easy to win. Note that after 10.Na5 white has the option of transposing to this line by playing 12.Nd6, rather than Nd8- that's why Na5 is more flexible. But white still has some work to do before he can queen in both the Nd6 and Nd8 lines.

Thanks for that info, it does help

I should mention that what the training tool does is let you play against the computer (rated 2000?) in a set position. So all of the variations I posted are the ones that the computer played against me. I just wish it came with explanations as to why your move was good/bad.