Snakes and Adders - A New Grid Puzzle

Question

_{So, literally last night, I dreamt of this new grid puzzle genre, so I'm just giving it a shot...}

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Rules of Snake and Adders:

1. Put some snakes into the grid on the white cells.
2. Each snake forms a sequential chain of cells connected horizontally and vertically from its head with '1' to its tail with 'N' where N is the length of the snake.
3. The length of each snake must be inside the given range on the top-right corner.
4. Snakes cannot touch other snakes horizontally or vertically, but diagonally is allowed. However, snakes can touch themselves.
5. The clue denotes the sum of the snake numbers adjacent to it including diagonals.

Answer 1

The complete grid

If we first look at this cell

since it's a high number and only 4 surrounding cells.

we see that we get three potential snake-formations, with sum 14, surrounding that cell.

OR

OR

Now, if we focus on this cell

it becomes clear that the only formation (in the previous case) that works is the second one.

This leads to:

Now we can fill in some more numbers

Let's continue following the rules and we get

and finally: