Sale of Second Home with Life Tenancy Rights

My mother purchased a townhouse in 2000 for $100,000.  In 2005, she "sold" me the townhouse for $1.00 (which, apparently in IRS terms, is considered a gift), but on the deed she retained life tenancy rights to remain in the townhouse and continue to use as her primary residence.   Note that I have my own home so  the townhouse would be considered a second home and not my primary residence.  In 2024,  my mother moved from the townhouse to a nursing facility and I sold the townhouse for $150,000, of which I received $135,000 as the remainderman and my mother received $15,000 as her portion of life tenancy rights.   I received a 1099-S listing the $135, 000 proceeds paid to me, but my mother did not receive a 1099-S.  For tax purposes, is my cost basis the purchase price my mother originally paid for the townhouse ($100,000), resulting in a capital gain of $35,000 to be reported on Schedule D on my taxes ($135,000 - $100,000)?  And does my mother need to report the $15,000 portion of proceeds on her taxes, and, if so, on which schedule and line should it be reported (keeping in mind it has been her primary residence since 2000)?   Thank you.