#6: Cracks on Glass (Sudoku variant)

I was really big on Sudoku-based puzzles back then, huh.

Rules

Classic 6x6 Sudoku rules apply. Additionally, connect pairs of orthogonally adjacent cells with straight lines between their centers, so that all cells form one connected network. Also:

• Cells with digits 1 and 2 should be directly connected to 1 neighbour each.
• Cells with digits 3 and 4 should be directly connected to 2 neighbours each.
• Cells with digits 5 and 6 should be directly connected to 3 neighbours each.

Here is an example of a valid grid with lines drawn in:

Main puzzle

Pretty difficult puzzle. Hint for break-in (select to make visible): [Consider the total number of lines against the number of cells. How many loops can the network have?]

Difficulty (select to make visible): ★★★⯪☆

Solve in Penpa

Solve in SudokuPad (answer check only for numbers)