I have a simpler proof.

look the graph first,

1

11

2 1

1 2 11

111 22 1

3 1 22 11

1 3 11 222 1

111 3 2 1 3 2 11

3 11 3 1 2 111 3 1 22 1

1 3 2 11 3 111 2 3 11 3 11 22 11

1 => 11

2 => 1 and 2

3 => 1 and 3

So, single 1 or 2 or 3 or 4(if exists) can never produce bigger number than itself. This is TRANSFORM A.

The only way to get bigger number is merging multiple continuous same numbers into one.

11 => 2 and 1

111 => 3 and 1

22 => 2 and 2

222 => 3 and 2

...

This is TRANSFORM B.

TRANSFORM A + TRANSFROM A, at best, we can get three 1s in a row. 1 2 => 11 12.

TRANSFROM B + TRANSFORM B, at best, we can get two legal numbers in a row. 222 33 => 32 23.

TRANSFROM A + TRANSFROM B, at best, we can get two legal numbers in a row. 2 11 => 12 21.

TRANSFROM B + TRANSFROM A, 111 2 => 31 12.

Since we only care multiple continuous same numbers, it is safe to generalize A + B cases shown above to A + B + C ... infinite case.

We can never produce 3 same numbers in a row.