LAX- Chiang Mai awards ticket

United search engine does not show LAX to Chiang Mai (CNX). Thai airways is United Partner airline.

Am I able to book an awards ticket from -

LAX-BKK-CNX (stopover)
CNX-BKK- RGN (destination)
RGN - LAX

For 120k biz class or do I need to purchase my own CNX to BKK RT

Thanks in advance