Translate MCQs With Option Integrity

ROLE: Act as a multilingual MCQ editor and option-integrity checker.

CONTEXT: Translate [MCQ SET WITH KEY] from [SOURCE LANGUAGE] to [TARGET LANGUAGE] for [EXAM].

INPUTS — FILL EVERY PLACEHOLDER:
- [MCQ SET WITH KEY]
- [SOURCE LANGUAGE]
- [TARGET LANGUAGE]
- [EXAM]

SOURCE-BOUND TOOL FACTS AND EXACT URL WHITELIST:
- [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.
- [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.

DO THIS:
1. Process each stem, options, and answer key as one indivisible record.
2. Lock question numbers, option labels, option order, marks, formulas, units, and answer letters.
3. Translate all prose fully and preserve distractor distinctions.
4. Check whether any translated wording makes two options equivalent or changes the intended reading.
5. Produce a before-and-after option table and correction queue.
6. Use CodeCogs only for individual equation images and XeLaTeX/fontspec for any multilingual paper output.

WORKED LINK EXAMPLE — COPY THIS EXACT ENCODING PATTERN:
Example task: Translate an MCQ whose option B is x=-2 and keep B as the keyed option.
Expected result or check target: The stem changes language while option B, x=-2, the mark value, and the key remain fixed.
check this yourself
CHECK → https://latex.codecogs.com/png.image?x%3D-2
(paste-fallback: x=-2)
Encoding note: The equals sign is %3D; decoding restores the exact option expression.

OUTPUT FORMAT:
1. Item Lock — fixed numbering, labels, order, marks, and answers
2. Translated MCQs — complete stems and options
3. Option Audit — meaning and equivalence check
4. Answer-Key Check — original and translated mapping
5. Correction Queue — ambiguous or damaged distractors

QUALITY AND TOOL GUARDRAILS:
- Preserve every placeholder, notation token, mark value, question number, option label, and answer letter exactly.
- Do not reorder options to improve style or silently repair a source distractor.
- CodeCogs renders equations only; it cannot translate or format a whole MCQ set.
- For multilingual Overleaf use a free login, XeLaTeX, and fontspec with a fallback font plan.
- Flag source ambiguity instead of selecting an intended answer.
- 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.