HAISU (Room Count): An original grid-logic challenge

HAISU is a portmanteau of three Japanese words - 'hairu', to enter, 'su', number, and 'hausu', an English borrow word meaning house, of course.
Together, we get a meaning of 'enter number house', which I have roughly translated to English as 'Room Count'.

The rules are simple - draw a path from the O to the X, passing through every cell in the grid exactly once. The grid is divided into several rooms. When your path passes over a cell with the big number N, it must be the Nth time you have entered the room. If a room has a small number m in the top left corner, you must enter that room a total of m times. An example Haisu puzzle and its unique solution are shown below.

Hopefully this example puzzle clarifies the rules. Your actual challenge is this!

No guessing, no handwavy steps, just pure logic required to solve this puzzle!

Edit: A long awaited fix.

Answer

Here is what I came up with by adding lines 1 by 1 where I could prove it was 100%. Please forgive the atrocious paint display after all my redo and deletions.

bottom right : only way to hit the 2 on the second entrance while keeping an exit open to go up after.

bottom right : only way to hit every tiles without entering more than 3 times.

right : only way to hit the 1 on first entrance and the 2 on second entrance. Cannot loop any higher without blocking the final loop for the exit(takes quite a bit of thinking to see it; could have been left for later when it is more obvious)

Only way to hit the 1 on first entrance without dooming the tile to its right to eternal solitude.



Must touch the 1 on first time in and must only enter 2 times.

Follow the only possible path to allow the correct amount of entrances.

More logical deductions based on required number of entrances.