You can only extend a tourist visa once for 30 days. Unless I'm missing something.

Yes you can leave and come back but you need to get a re-entry permit from immigration or airport.

If you want 120 days out of a 60 day visa you will have e to leave the country anyway for a border bounce.

Your options are

60 visa

Extend 30 days in country 1900

Border bounce 30 day exempt.

Or

30 day visa exempt entry.

30 extension at immigration office.

Border bounce for another 30 day visa exempt.

Extend 30 days at immigration.

Your 90 days stops when you leave the country.

It restarts from 0 when you reenter.

There is no reason to do a 90 day report while you are out the country. Simple reason - you are out the country.

Don't forget your reentry permit if you are on a single entry.