Large language models (programs trained on text) often write out each step when solving a math problem, like a person working through a calculation. This 'chain of thought' is accurate but slow and costly. Newer approaches keep some steps inside the model as hidden computations, but that is less reliable for exact reasoning.
A method called VITAL decides when to write a token (a piece of a word) as explicit reasoning and when to keep the computation internal. A module learns which tokens are most important to write out. The model is trained on examples mixing written and internal steps, then fine-tuned with a reward for both correct and shorter answers.
On benchmarks, VITAL improved accuracy by 9.6% on one model and 5.1% on another, with similar step counts. Compared to other mixed approaches, it needed far fewer written tokens for higher accuracy. This shows selective verbalization reduces cost without harming correctness.