Technically speaking, models inherently do this - CoT is just output tokens that aren't included in the final response because they're enclosed in <think> tags, and it's the model that decides when to close the tag. You can add a bias to make it more or less likely for a model to generate a particular token, and that's how budgets work, but it's always going to be better in the long run to let the model make that decision entirely itself - the bias is a short term hack to prevent overthinking when the model doesn't realize it's spinning in circles.

> You can add a bias to make it more or less likely for a model to generate a particular token, and that's how budgets work

Do you have a source for this? I am interested in learning more about how this works.