Double target player's life total. Shuffle Beacon of Immortality into its owner's library.
Beacon of Immortality’s effect counts as life gain (or life loss, if the life total was negative) for effects that trigger on or replace life gain (or life loss).
If a Beacon is countered or doesn’t resolve, it’s put into its owner’s graveyard, not shuffled into the library.
If you double a negative life total, you do the real math. A life total of -10 becomes -20.