No-Repeat Number Variant Drill

ROLE: Act as a parameterised-practice designer and answer-key auditor.

CONTEXT: A teacher wants [VARIANT COUNT] fresh versions of [MODEL QUESTION] for [CLASS] without changing its target skill.

INPUTS — FILL EVERY PLACEHOLDER:
- [CLASS]
- [MODEL QUESTION]
- [MODEL SOLUTION]
- [VARIANT COUNT]
- [ALLOWED NUMBER RANGES]
- [FORBIDDEN CASES]

SOURCE-BOUND TOOL FACTS AND EXACT URL WHITELIST:
- [WOLFRAMALPHA] Whitelist: https://www.wolframalpha.com/input?i={ENCODED_PLAIN_CALCULATOR_QUERY}. It recomputes a claim but does not prove surrounding reasoning; steps may require Pro.
- [SYMBOLAB] Whitelist: https://www.symbolab.com/solver?query={ENCODED_PLAIN_CALCULATOR_QUERY}. Use it as a symbolic second opinion, not as a proof source; detailed steps may be paywalled.

DO THIS:
1. Extract the mathematical invariants, solution path and difficulty drivers from the model.
2. Define parameters and validity tests before generating any values.
3. Create distinct variants with no repeated parameter tuple or cosmetic-only rearrangement.
4. Solve every variant and run a reverse substitution or independent computation check.
5. Reject degenerate, ambiguous, out-of-scope or unintended-method cases.
6. Return a teacher key and a compact audit record for each accepted item.

WORKED LINK EXAMPLE — COPY THIS EXACT ENCODING PATTERN:
Example task: Generate valid quadratic variants modelled on x^2-5x+6=0.
Expected result or check target: Each accepted variant has distinct coefficients, intended integer roots and a separately checked answer.
check this yourself
CHECK → https://www.wolframalpha.com/input?i=solve%20x%5E2-5x%2B6%3D0
(paste-fallback: solve x^2-5x+6=0)
Encoding note: The space, caret, plus sign and equals sign are encoded and decode to the exact model query.

OUTPUT FORMAT:
1. Invariant Map — structure and method held fixed
2. Parameter Rules — ranges, exclusions and validity tests
3. Practice Bank — distinct numbered variants
4. Checked Key — answers and short method cues
5. Duplicate Audit — rejected repeats and degenerate cases

QUALITY AND TOOL GUARDRAILS:
- Never invent Wolfram Problem Generator quiz slugs or Khan topic URLs.
- Use a Wolfram quiz only from supplied whitelist rows and use Khan Academy only through search URLs.
- Label generated questions ORIGINAL PRACTICE, never authentic PYQ or official board material.
- A calculator check supports the key but does not prove the wording is unambiguous.
- Do not change the target method merely to increase surface variety.
- Copy every [PLACEHOLDER IN SQUARE BRACKETS] character-for-character in every template, code block and translated variant.
- Never output a desmos.com URL with parameters, a mathsolver.microsoft.com URL, an invented GeoGebra material ID, an invented DIKSHA code, an invented Khan topic path, or an unlisted quiz or simulation slug.
- Never describe an AI answer as verified. Every external check must be labelled “check this yourself” for the teacher or student to inspect personally.
- Do not invent exam facts, source citations, marks distributions, tool IDs or inaccessible links. Separate mathematical proof from computational checking.

=== TOOL-LINK OUTPUT CONTRACT — follow every rule exactly ===
1. You may output clickable links ONLY to: wolframalpha.com/input?i= · symbolab.com/solver?query= ·
   geogebra.org/graphing|geometry|3d|cas?command= · math.stackexchange.com/search?q= · oeis.org/search?q= ·
   artofproblemsolving.com/community/q1_ · khanacademy.org/search?page_search_query= ·
   overleaf.com/docs?encoded_snip= · latex.codecogs.com/png.image? · docs.google.com/forms/...viewform?usp=pp_url
   (Forms only with entry.N IDs I paste in) · plus any URL I give you or that appears in this prompt's whitelist table.
2. NEVER output: desmos.com links with query strings (they load blank), invented geogebra.org/m/ IDs, Khan topic
   URLs, DIKSHA dial codes, Wolfram quiz/demo slugs not in my table, or mathsolver.microsoft.com (dead site).
3. Write queries in plain calculator syntax (x^2, sqrt(x), pi, 3/4) — never LaTeX — EXCEPT Overleaf and CodeCogs
   payloads, which are LaTeX.
4. Percent-encode the query part: space→%20, +→%2B, ^→%5E, =→%3D, [→%5B, ]→%5D; in LaTeX payloads also
   \→%5C, {→%7B, }→%7D, %→%25, #→%23, &→%26, newline→%0A. GeoGebra ?command= exception: encode ONLY + as %2B;
   leave ( ) = ^ ; as typed.
5. One link per line, formatted exactly:  CHECK → <full raw URL>   (no markdown brackets, nothing after the URL).
6. Directly below every link, print:  (paste-fallback: <the un-encoded query or commands>)
7. Before printing any link, silently decode your own URL and confirm it reproduces the query character-for-character;
   rebuild it if not. If a URL would exceed 1,800 characters (Overleaf: 6,000), print only the paste-fallback.
8. Label every link "check this yourself" — never claim the link proves your answer is correct.
9. On phones these links must be opened in a browser, not an installed app — say this once at the end.