Advanced pawn endgame

I've never before seen a pawn endgame with this many pawns! I focused on advancing my king and my majority before my opponent could promote, although Stockfish says the final position is drawn!

The final position may well be a draw indeed. Your King is more active, but he still has one more Pawn. You win the Pawn back, but in the mean time he can activate his King. For example
36 Ke4 Kg5 37 Kd5 Kf4 38 Kxc5 Ke3 39 Kxb5 Kd2

Want to see some instructive pawn endgames with a lot of pawns ?
Look at 4 GMs EG here -> www.dropbox.com/s/jpp4l8jdtbl5bai/Pawns%20endgame%20-%205%20GM%20games.docx?dl=0

And somme collections on chessgames :
http://www.chessgames.com/perl/chesscollection?cid=1018188
http://www.chessgames.com/perl/chesscollection?cid=1006053
http://www.chessgames.com/perl/chesscollection?cid=1013499

And some more general advices here : http://www.chessgames.com/perl/chesscollection?cid=1007134

In my opinion, questionable black to move 26. ... f6!? and even worse move 27. ... gxf? - allows you to create white an outside passed pawn ( I do not know whether I translate into English). And broken draw. There is a saying in the English language?

The English expression is a distant passed pawn, but outside is clear enough.
I doubt 26...f6 or 27...gxf6 are wrong. Sure, white gets the distant passed pawn, but the game seems a draw anyway.
What else could Black have done?

@tpr Instead of running the king to protect the pawn g5 it was necessary to send a pawn d5 in Queens.

You mean instead of 31...Kf6 31...d4? Then 32 Kxg5 c4 33 Kf4 and white stops the d-pawn with his king, while the black king must stay with the h-pawn.
I think 31...Kf6 was right and leads to a draw.

I agree with @tpr , both players played accurately and the final position is drawn, despite White's king being so dominant.

Also I believe that had White pushed anything on the queenside (maybe other than a4) that would only have made the pawns more vulnerable and put White at a disadvantage!