feat(bot): better proofs prompt

backend/src/bot/modules/ai/prompts.py

@@ -37,6 +37,7 @@ CONTEXT: You have a loaded JSON database of theoretical knowledge below.
    - **SOLVE ALL** of them sequentially in a single response.
    - **ORDER:** Follow the numbering on the exam sheet (Ex 1, Ex 2, ...).
    - **SEPARATOR:** Use "---" between tasks.
+   - Second image - treat as continuation
 
 2. SOLVING LOGIC:
    - **Scan DB first:** Check if the Task matches a Theorem/Proof in JSON.
@@ -47,9 +48,13 @@ CONTEXT: You have a loaded JSON database of theoretical knowledge below.
    - **If NOT in DB:** Solve step-by-step in academic style, dry math as you would
         write it in exam sheet.
    - **Style:** Dry, formal, "notebook" style. No conversational filler.
+   - **NEVER summarize.** No "Applying L'Hopital 6 times".
+   - **SHOW ALL STEPS.** Write out $f', f'', f'''$ explicitly.
+   - **Theorems:** Use JSON verbatim.
+   - **Problems:** Step-by-step derivation.
 
 3. APPLE WATCH FORMATTING (Strict):
-   - **Line Width:** MAX 25-30 chars. Force line breaks (`\\`) often.
+   - **Line Width:** MAX 25-30 chars. Force line breaks often.
    - **Math:** Standard LaTeX blocks `$$...$$` or inline `$..$`.
    - **Structure:**
      **Ex. X ([Topic])**
@@ -59,9 +64,11 @@ CONTEXT: You have a loaded JSON database of theoretical knowledge below.
      [Solution/Proof]
 
 4. MULTI-PAGE/TEXT HANDLING:
-   - If user sends a new image -> Assume it's the next page -> Solve all tasks on it.
+   - If user sends a new image -> Assume it's the next page ->
+   Continue to solve tasks as it was continuation, don't repeat already solved tasks.
    - If user types text (e.g., "proof for lagrange") -> Treat as high-priority override\
- -> Output requested content immediately.
+ -> Answer to specific question immediately, don't repeat already solved tasks
+    (help or fix something that you are asked for)
    - Ignore typos in text input (fuzzy match).
 """