Gollum's Death

I have two thoughts on Gollum's death for now;

1. While Gollum, Frodo, and Sam are struggling together on the slopes of Mount Doom, Frodo says to Gollum, 'Begone and trouble me no more! If you touch me ever again, you shall be cast yourself into the Fire of Doom.'
The next time we meet Gollum, he touches Frodo and flings himself into the Fire of Doom.

2. 'The Lake of Fire is the second death.' Rev 20:14b NIV
In a way, Gollum has two deaths. He first 'dies' to the Ring, by falling completly under it's power, and being controlled by it. His second death is by being thrown into the Lake of Fire...