common/parser: add proper reasoning tag prefill reading (#20424)

* Implement proper prefill extraction

* Refactor cli parameters, update docs, move reasoning budget sampler part to common/reasoning-budget.cpp

* Update tools/server/server-task.cpp

* refactor: move grammars to variant, remove grammar_external, handle exception internally

* Make code less C++y

Co-authored-by: Georgi Gerganov <ggerganov@gmail.com>

tools/server/README.md

@@ -907,7 +907,7 @@ If query param `?fail_on_no_slot=1` is set, this endpoint will respond with stat
       "chat_format": "GPT-OSS",
       "reasoning_format": "none",
       "reasoning_in_content": false,
-      "thinking_forced_open": false,
+      "generation_prompt": "",
       "samplers": [
         "penalties",
         "dry",
@@ -972,7 +972,7 @@ If query param `?fail_on_no_slot=1` is set, this endpoint will respond with stat
       "chat_format": "GPT-OSS",
       "reasoning_format": "none",
       "reasoning_in_content": false,
-      "thinking_forced_open": false,
+      "generation_prompt": "",
       "samplers": [
         "penalties",
         "dry",
@@ -1193,7 +1193,7 @@ The `response_format` parameter supports both plain JSON output (e.g. `{"type":
 
 `reasoning_format`: The reasoning format to be parsed. If set to `none`, it will output the raw generated text.
 
-`thinking_forced_open`: Force a reasoning model to always output the reasoning. Only works on certain models.
+`generation_prompt`: The generation prompt that was prefilled in by the template. Prepended to model output before parsing.
 
 `parse_tool_calls`: Whether to parse the generated tool call.