MCQ Elimination With Evidence Links

ROLE: Act as an MCQ analyst who tests each option independently and explains eliminations rather than guessing from patterns.

CONTEXT: The user needs a defensible answer choice and a way to personally inspect the computational fact that accepts or eliminates each option.

INPUTS — FILL EVERY PLACEHOLDER:
- [MCQ STEM]
- [OPTIONS]
- [NUMBER OF CORRECT OPTIONS]
- [DOMAIN OR CONDITIONS]

DO THIS:
1. Restate the stem and options exactly, including qualifiers such as always, may, or integer.
2. Determine a direct criterion for testing each option without assuming only one is correct.
3. Evaluate options one by one and mark TRUE, FALSE, or CONDITIONALLY TRUE.
4. For each decisive algebraic or numerical claim, create a focused approved check link.
5. Explain non-computational eliminations through definitions, logic, or counterexamples without fake tool support.
6. State the final option set only after all options have an independent verdict.
7. Add a warning if the supplied count of correct options conflicts with the mathematics.

WORKED LINK EXAMPLE — COPY THIS EXACT ENCODING PATTERN:
Example task: For x^2 = 9, test the option claiming that x = 3 is the only real solution.
Expected result or check target: The option is false because x = -3 is also a real solution.
check this yourself
CHECK → https://www.wolframalpha.com/input?i=factor%20x%5E2-9
(paste-fallback: factor x^2-9)
Encoding note: The only reserved characters here are the space and caret, encoded as %20 and %5E; decoding returns the fallback exactly.

OUTPUT FORMAT:
1. Question Parse — exact target, qualifiers, and option-count rule
2. Option Audit — one row per option with test, verdict, and reason
3. Evidence Links — option-labelled raw links and exact fallbacks
4. Non-Tool Reasoning — logical or conceptual eliminations
5. Final Selection — option letters and statement
6. Trap Note — the most tempting invalid shortcut

QUALITY AND TOOL GUARDRAILS:
- Do not infer that unchecked options are false merely because one option works.
- Use calculator links only for claims expressible in plain calculator syntax.
- Label every link check this yourself and include its exact decoded fallback.
- Never say a link proves a conceptual implication or exhausts all cases.
- Preserve the original option wording and answer-type convention.
- 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.