Do you think this will run well with Crysis 3?

Howdy. I'm going for an upgrade to my graphics card, because mine simply cannot play Crysis 3.
Based on my system specs:
CPU: AMD FX-4100 3.60Ghz
MoBo: Gigabyte M68MT-S2 with a NVIDIA GeForce 7025 chipset / nForce 630a (southbridge)
RAM: 8GB DDR3
Graphics: AMD Radeon HD 6670
PSU: TurboLink 500W-BK 12v

RECOMMENDED SYSTEM OPERATING REQUIREMENTS FOR PC:
• Windows Vista, Windows 7 or Windows 8
• DirectX 11 graphics card with 1GB Video RAM
• Quad core CPU
• 4GB Memory
Example 1 (Nvidia/Intel):
• Nvidia GTX 560
• Intel Core i3-530
Example 2 (AMD):
• AMD Radeon HD5870
• AMD Phenom II X2 565

HI-PERFORMANCE PC SPECIFICATIONS:
• Windows Vista, Windows 7 or Windows 8
• Latest DirectX 11 graphics card
• Latest quad core CPU
• 8GB Memory
Example 1 (Nvidia/Intel):
• NVidia GTX 680
• Intel Core i7-2600k
Example 2 (AMD):
• AMD Radeon HD7970
• AMD Bulldozer FX4150

Does anyone have a good idea on what type of card I should get for my build? If possible I don't want to go over $250..
Thank you!