GPU rendering not working

Windows 7 ATI Radeon HD5570
I have the latest driver installed.
When I chose GPU in the preferences System panel, my choice is “Redwood”. In the render panel, if I choose “GPU Compute” I can choose “Supported”. When I render, I get a message (follows the normal info about frame, time, etc.) that it is loading render kernels and may take a few minutes the first time. Then, nothing actually renders. There are no error messages, and the render completes (progress goes to 100%), but there just isn’t any rendered image. Just a gray and black checkerboard.

To quote from the manual:
“OpenCL is supported for GPU rendering with AMD graphics cards. We only support graphics cards with GCN architecture (HD 7xxx and above).”

so it seems I would need a new GPU. Has anyone discussed which GPU card might be a good choice for use with Blender? I don’t care about gaming, just cycles rendering. I don’t want to spend $1000, but maybe $200-300 is OK.

The question should rather be: Has anyone not discussed this?
The “Technical Support” section is more or less full of threads about this very topic.