Puzzle of the week #277

Black has sacrificed a piece to reach this position where it threatens mate in one (1... Qg2#) or mate in two (1.Rg1 Qxh2+ 2.Kxg2 Rh8#). Your tasks:
a) Decide what should white do here: resign or play ahead?
b) If you chose play ahead, show white's best response you can think of
Explaining your solution gets you extra points! Total available points for this puzzle is 22. The answers will be published next time together with puzzle #278.

Puzzle #276 solution:
Game Schiffers – Steinitz, Rostov 1896. Benjamin's answer:
"a) Material: white is up 2 points
King safety:
- both kings have castled
- both kings are exposed
Center:
- white's bishop controls e5 and d4
- white's queen controls e4
- white's c4 pawn controls d5
- black's queen controls d4
- black's c6 pawn controls d5
- blacks bishop controls d5
Pieces:
- white has double rooks
- the white rooks attack the black bishop
- black is pinning the bishop to the queen
- the black d3 pawn attacks the white queen
- the black queen attacks the f2 rook
- the black knight attacks the white bishop
Conclusion: I think white is better
[Event "Puzzle #276"][Date "2015.04.20"][Result "1-0"][SetUp "1"][FEN "4r1k1/pp1n1b2/1qp3p1/4B1P1/P1P4P/3p4/2P1QR1K/5R2 w - - 0 1"]1.Rxf7 dxe2 2.Rg7+ Kh8 3.Rxd7+ Kg8 4.Rg7+ Kh8 5.Re7+ Kg8 6.Rxe8+ Kh7 7.Rh8# (7.Rf7#)

Correct solutions:
Hyson, Benjamin, Jalen - 20 points
Coco, Cody - 18 points
Yakov, Bradley - 14 points
Aaron, Deryk - 12 points
Uros - 10 points
Terry, Dheera - 2 points
Andrew - 1 point

STANDINGS:
Coco - 188 points
Bradley - 165 points
Cody - 163 points
Jalen - 154 points
Benjamin - 152 points
Hyson - 145 points
Deryk - 113 points
Aaron, Yakov - 102 points
Terry - 69 points
Uros - 25 point
Dheera - 9 points
Andrew - 8 points