Rules vs Narrative System

Rules Dependent: There's a fire burning behind you and you're on this narrow ledge and in front of you is this (hopefully) not too wide Chasm of Certain Death. According to my character sheet I should be able to do X, Y, or Z. Y gives me the best modifiers up front. "OK GM I'll Y, [roll the dice...]." And according to the rules you either succeed or fail, and the GM has to decide - according to the rules - what fail looks like (usually a certain amount of damage as you plunge to your doom).

Narrative Dependent: There's a fire burning behind you and you're on this narrow ledge and in front of you is this (hopefully) not too wide Chasm of Certain Death. "OK GM, I'll take a really deep breath, focus on that handhold I think I see on the other side that should help pull me up or over and then I'll jump..."

The player really just narrated two actions: Create Advantage to Notice a better way to survive the jump and Overcome using Athletics to make the jump. The player rolls Notice and spots the handhold he wants (1 invoke). The GM states the fire has gotten closer, the player rolls Athletics - and gets a case of the uglies with no Fate Points handy (or that he's willing to spend). The first roll is -3, the reroll (from the I See a Handhold invoke) is -3. He fails by 2: Fail (not an option = falling into Chasm of Certain Death) or Succeed At Cost.

The Player suggests "I jump and... miss the other side but fall onto a ledge that I hadn't seen below which is leading to another opening in the chasm face and further into the caverns." The GM agrees, and states because he failed the roll he takes 2 points of stress and the story moves on in a direction neither of them expected...

That GM can now either have our character meet a Slumbering Giant Salamander (to take advantage of the fact that he has no FP and a 2 stress hit), he can toss in either a skill check or a challenge which has him squirming through Tight Torturous Spaces (which, at a high enough difficulty could cause additional stress) to get back to the originally planned scene, or he can simply say that the character finds the Stairway Going Up leading back to the originally planned scene, routing him right back into the original storyline.

The player can take advantage of the situation by noting that his character has a stunt called Unusually (and Unexpectedly) Lucky which allows him to drop a Situation Aspect with 1 invoke for free - and he injects the idea that there's a Map and a Key to a Massive Lock lying next to a skeleton down there.

Heroes in movies or books rarely fail and when they do it's a fail forward  in a way that progresses the story even if, temporarily or permanently, it deviates from the original storyline. This is the Fate Way.