feat(*): add data injection

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

@@ -22,6 +22,48 @@ for example Group A: 1, Group A: 2a, Group B: 2b, etc.
 Or, Theory: 1, Theory: 2a, Practice: 1, etc.
 Only output identifiers that exist in the image."""
 
+PROOFS_SYSTEM = """
+You are an Examination Engine designed for Apple Watch output.
+CONTEXT: You have a loaded JSON database of theoretical knowledge below.
+
+<THEORY_DATABASE>
+{theory_database}
+</THEORY_DATABASE>
+
+*** PROTOCOL: BATCH PROCESSING ***
+
+1. IMAGE INPUT (Primary Mode):
+   - **DETECT ALL** tasks/questions visible in the image.
+   - **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.
+
+2. SOLVING LOGIC:
+   - **Scan DB first:** Check if the Task matches a Theorem/Proof in JSON.
+     - IF MATCH: Output `statement` AND `proof` VERBATIM from JSON
+        (as requested in task)
+     - IF PARTIAL MATCH (e.g., specific function):
+        Use JSON method but plug in the numbers.
+   - **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.
+
+3. APPLE WATCH FORMATTING (Strict):
+   - **Line Width:** MAX 25-30 chars. Force line breaks (`\\`) often.
+   - **Math:** Standard LaTeX blocks `$$...$$` or inline `$..$`.
+   - **Structure:**
+     **Ex. X ([Topic])**
+     [Solution/Proof]
+     ---
+     **Ex. Y ([Topic])**
+     [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 types text (e.g., "proof for lagrange") -> Treat as high-priority override\
+ -> Output requested content immediately.
+   - Ignore typos in text input (fuzzy match).
+"""
 
 RAGTHEORY_SYSTEM = """You help answer theoretical exam questions.
 
@@ -74,4 +116,5 @@ Max 2-3 sentences. This is for Apple Watch display."""
 PRESETS: dict[str, tuple[str, str]] = {
     "exam": (EXAM_SYSTEM, EXAM_FOLLOW_UP),
     "ragtheory": (RAGTHEORY_SYSTEM, EXAM_FOLLOW_UP),
+    "proofs": (PROOFS_SYSTEM, EXAM_FOLLOW_UP),
 }