To extend what is stated in Die4ever's all puzzles tutorial:

Short: We use a reset-strat to reset until certain conditions are met, namely, until it's possible to encase the goal square with a ^ piece above and a v piece below (Crab https://imgur.com/a/mYIjh5Q ). If that is satisfied you have a very high likelihood of a 3 move win, by repeating that instruction with the next v or ^ you get, and then putting any piece in the other slot, forming a zig-zag path from the start to the goal.

Long: For any randomly given game there's probably a chance between 50 and 70 % that you'll win. That requires making the perfect move at each step though, and it'll be a long game and you're still not guaranteed to win. Therefore, we reset until we have a shot at a SHORT game we can easily win.

The steps are explained in REVERSE, for reasons that will become obvious. We will call this position the crab: https://imgur.com/a/mYIjh5Q And we will call the piece that pops out on the left, which you have to insert, the "free piece".

Winning Move: Given that there are no straight pieces, the shortest possible way from the center to the goal on the right is therefore either a zigzag in the shape of

^v^ going northeast, southeast northeast https://imgur.com/a/pLhx8oj or v^v going southeast, northeast, southeast https://imgur.com/a/Z47j3Dh

If you have that, and the mouse is on the center (or at least connected to the center) you win.

1 Move Away: The way to get to that position is by pushing any piece in a slot next to the goal that will establish that winning position. The position before that looks like one of these 2 pictures [only the 3 tiles next to the goal are important]:

The Crab with another v west to the goal: https://imgur.com/a/04kTcn6 Whatever piece you put where the ^ currently is establishes the ^v^

The Crab with another ^ west to the goal: https://imgur.com/a/K17xwvl Whatever piece you put where the v currently is establishes the v^v

2 Moves away: There are 2 ways to get to the 1-move-away position.

Way 1: Starting with the Crab and either ^ or v as your free piece. Putting a ^ in the top position or a v in the bottom position will bring you to the shape of 1-move-away

Way 2: Starting with the goal encased in two ^ ^ pieces and a v as a free piece, or Starting with the goal encased in two v v pieces and a ^ as a free piece. For an example of the latter, see here: https://imgur.com/a/9g7Wfyp In this case, use your free piece to form the crab (by putting the ^ above or the v below). Then, you're 1 move away.

3 Moves away: There are 3 ways to get to the 2-moves-away scenario, one of which is unlikely.

Way 1: The crab is there, but you don't have ^ or v as a free piece. In this case you can recycle your piece somewhere that doesn't interfere with the crab, and wait for the next piece.

Way 2: The Crab is missing one arm, and the one it's missing is your free piece: complete the Crab!

Way 3 (unlikely): The crab is missing one arm, and you have a copy of the arm it has as a free piece. Put that copy in the other slot and hope for the third correct piece to come. Since you now need one specific piece, you have a vaguely 1/4 chance of getting that, whereas the crab can continue with 2/4 pieces. Therefore this way is less likely to succeed.

In every other situation, reset. It is possible that Stauf will thwart your plans with his move. You cannot "undo" one of his moves by going the opposite direction of where he went. Usually this requires resetting as well.

Additional discovery by Jobee:

If you start with a double v and have a caret, or a double caret and have a v (as in this screenshot https://imgur.com/a/Vr77W0S ), completeing the crab will already put you one move away: https://imgur.com/a/juPoYy7 and you can finish with any tile