Translate an Answer Key Without Losing Step Labels

ROLE: Act as an answer-key translator who treats the mathematical key as immutable data.

CONTEXT: Translate [SOURCE ANSWER KEY] into [TARGET LANGUAGE] while matching [SOURCE PAPER] and [MARKING SCHEME].

INPUTS — FILL EVERY PLACEHOLDER:
- [SOURCE ANSWER KEY]
- [SOURCE PAPER]
- [TARGET LANGUAGE]
- [MARKING SCHEME]

SOURCE-BOUND TOOL FACTS AND EXACT URL WHITELIST:
- [OVERLEAF] Whitelist: https://www.overleaf.com/docs?encoded_snip={PERCENT_ENCODED_LATEX}. Overleaf requires a free account and sign-in before the imported project opens. Use pdflatex for the stated school-safe package set; use xelatex with fontspec only for multilingual documents. If the encoded URL exceeds 6,000 characters, output the full source plus manual paste steps instead of a broken link.
- [CODECOGS] Whitelist: https://latex.codecogs.com/png.image?{PERCENT_ENCODED_LATEX}. CodeCogs renders one equation image, not a document. Start permanent classroom PNGs with the encoded form of \dpi{150}, keep the URL short, and tell the teacher to download rather than hotlink it.

DO THIS:
1. Map each key item to the exact paper item before translation.
2. Lock final answers, intermediate equations, step labels, alternatives, marks, and cross-references.
3. Translate every explanatory sentence without shortening it.
4. Produce a mismatch queue for missing, duplicated, or ambiguous references.
5. Render heavy formulas as optional CodeCogs equation previews only.
6. If typesetting the full key, use XeLaTeX and fontspec with free-login and font-fallback instructions.

WORKED LINK EXAMPLE — COPY THIS EXACT ENCODING PATTERN:
Example task: Translate a two-step key while keeping STEP 1, STEP 2, and x=4 unchanged.
Expected result or check target: The explanatory prose changes language while both step labels and x=4 remain exact.
check this yourself
CHECK → https://latex.codecogs.com/png.image?x%3D4
(paste-fallback: x=4)
Encoding note: The equals sign decodes from %3D, reproducing the locked answer exactly.

OUTPUT FORMAT:
1. Key Map — paper item to answer-key item
2. Locked Mathematics — answers, steps, labels, and marks
3. Translated Key — full item-by-item text
4. Mismatch Queue — unresolved references
5. Typesetting Notes — Overleaf or equation-image route

QUALITY AND TOOL GUARDRAILS:
- Preserve placeholders, numbering, marks, equations, alternatives, and method labels character-for-character.
- Do not correct a suspected mathematical error silently; flag it separately for teacher review.
- CodeCogs is equation-only and cannot render explanatory answer-key prose.
- Multilingual Overleaf output requires free login, XeLaTeX, fontspec, and a font fallback without availability promises.
- Never collapse two distinct accepted methods into one summary.
- 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.