G0: conversation_scaffolding — Chat-format metadata and message-boundary tokens that frame the assistant’s response lifecycle rather than mathematical content.

C13: analysis_to_final_handoff — Boundary tokens that close internal reasoning and open the assistant’s final answer message.

C41: runtime_metadata_date — System-preamble metadata centered on the current date and other runtime header fields.

C46: system_preamble_identity — The system-header identity text describing the assistant and knowledge cutoff.

G1: local_conclusions — Short commitment moves that state a resolved value, confirm a candidate, or pivot from solved reasoning into answer delivery.

C12: answer_value_commitment — A short clause that finishes a numerical derivation by directly stating the resolved value or option.

C18: verification_closure — Brief confirming language that says a candidate works, a maximum is attained, or a special case checks out.

C45: answer_drafting_transition — Metadiscursive language about giving the explanation or crafting the final response once the reasoning is settled.

G2: final_solution_exposition — Polished final-answer prose where the assistant explains results step by step, states boxed conclusions, and writes pedagogical solution text.

C0: displayed_result_line — Standalone displayed result tokens such as a computed number or boxed quantity right before explanatory prose continues.

C4: applied_numeric_steps — Concrete arithmetic in final solutions for percentages, times, totals, discounts, and other everyday numerical computations.

C14: formal_derived_relations — More theorem-like final exposition that states derived relations, lemmas, or structured proof equations.

C16: final_answer_banner — Markdown-heavy final-answer openers like “Answer:” or “Solution” that launch the polished response.

C23: aligned_equation_solving — Dense, aligned symbolic derivations in final prose that solve systems or transform equations step by step.

C26: implication_chain_algebra — Arrow-style derivations that push a condition through successive logical-algebraic implications.

C27: boxed_choice_conclusions — Compact final statements that present the selected option, interval, axis, or boxed parameter value.

C31: formula_substitution_block — Plugging known quantities into a standard formula and then carrying out the resulting expression in final exposition.

C39: definition_and_classification — Final-solution exposition that teaches definitions, categories, or conceptual cases before applying them.

C47: worked_example_explanation — Long pedagogical answer blocks for straightforward quantitative problems, usually with numbered steps.

C49: inline_symbolic_simplification — Mid-exposition algebra or trig simplification embedded directly inside the polished solution narrative.

C56: tabular_and_case_data — Table-driven or list-driven final exposition that reads off inverse values, remainders, exponents, or enumerated case data.

G3: quantitative_word_problem_arithmetic — Rate, ratio, percentage, and unit computations used to turn verbal quantities into concrete numerical answers.

C21: rate_equation_setup — Setting up work-rate, speed, or shared-quantity equations from the verbal structure of a word problem.

C42: unit_conversion_division — Explicit division or conversion steps that turn totals into percentages, times, speeds, or per-unit quantities.

G4: problem_restatement_and_setup — Ingress tokens where the task, givens, options, and initial algebraic setup are restated before deeper reasoning begins.

C1: formal_problem_conditions — Restated functional equations, recurrence data, and algebraic premises that specify the mathematical object under analysis.

C7: option_listings — Multiple-choice and proposition blocks that enumerate answer options or statement sets before analysis begins.

C15: plain_english_problem_paraphrase — Rephrasing a word problem in simpler language to pin down what the quantities mean.

C20: analysis_entry_stub — Generic “We need to…” analysis openers that bridge from the prompt into concrete setup.

C44: algebraic_reexpression_setup — Initial substitutions and algebraic re-expression moves that translate givens into a workable symbolic form.

C51: user_problem_opening — The literal start of the user’s problem statement, especially “Problem: …” openings.

C53: full_geometry_prompt_body — Long-form geometry prompt text with named points, conics, tangencies, and target claims.

G5: geometric_model_construction — Geometry-specific setup that builds diagrams into coordinates, angle relations, vector conditions, and proof targets.

C10: standard_geometry_identities — Invoking law-of-sines, law-of-cosines, and other stock triangle identities to relate sides and angles.

C25: angle_chase_exploration — Geometry reasoning centered on chasing equal angles in isosceles, cyclic, or perpendicular configurations.

C40: diagram_sanity_checks — Geometry reasoning that questions whether a diagram interpretation, tangent statement, or projection description is even coherent.

C43: coordinate_frame_setup — Choosing axes, centers, foci, or canonical coordinates to model a geometry or conic problem.

C48: proof_goal_focusing — Sentences that explicitly state the geometric target—midpoint, locus, intersection, or quantity—to guide the next proof move.

C54: geometry_givens_unpacking — Expanding a geometry statement into the concrete objects, perpendiculars, tangencies, and known lengths needed for setup.

C55: canonical_figure_coordinates — Placing standard figures such as squares, circles, arcs, or regular polygons into convenient coordinates.

G6: combinatorial_structure_counting — Discrete-structure reasoning that counts arrangements or derives extremal constraints from configuration-level combinatorics.

C22: local_constraint_arguments — Configuration-based combinatorial reasoning about what row, vertex, neighbor, or assignment patterns are forbidden or forced.

C28: direct_arrangement_counting — Counting discrete arrangements, partitions, or selections by mapping the setup to a direct combinatorial formula.

G7: exploratory_search_and_disambiguation — Tentative reasoning used to test interpretations, search for the right lemma or representation, and resolve ambiguities mid-solution.

C2: geometry_theorem_search — Tentative geometry-proof moves that hunt for the right cyclic, midpoint, or angle lemma before a proof path is fixed.

C30: spatial_configuration_interpretation — Exploratory parsing of physical layouts, foldings, colorings, dice stacks, or other spatial-combinatorial setups.

C33: abstract_extremal_reframing — Recasting a discrete problem into a graph, matching, coloring, or worst-case formulation.

C35: known_formula_retrieval — Calling up a standard counting or game formula, then adapting it while still exploring whether it fits.

C36: auxiliary_variable_introduction — Introducing substitute variables, functions, or parameters as an exploratory route to simplify a hard expression.

C38: error_checking_and_reinterpretation — Backtracking over a possible misread, rounding issue, or mistaken remainder to reinterpret the problem correctly.

C57: wording_disambiguation — Reasoning focused on what an ambiguous phrase in the problem statement most likely means.

G8: case_analysis_and_condition_checking — Reasoning that partitions by domain, sign, monotonicity, or special cases to validate when statements or solutions hold.

C5: counterexample_statement_testing — Checking whether a claim is always true by probing edge cases, options, or counterexamples.

C9: solution_count_conditions — Determining when equations or set definitions have zero, one, or multiple valid solutions.

C11: symmetry_and_congruence_reduction — Using evenness, periodicity, modular equivalence, or structural symmetry to simplify the condition being checked.

C50: monotonicity_derivative_analysis — Checking where a function increases, decreases, or attains an extremum via derivatives or sign changes.

C58: interval_case_split — Describing the relevant domain or interval pieces and solving separately on each region.

C59: root_and_sign_feasibility — Checking zeros, positivity, evenness, or other feasibility conditions that determine whether a case is admissible.

C62: divisibility_bound_filtering — Applying divisibility, inequality, or bound filters to narrow a candidate set to the feasible cases.

G9: explicit_enumeration_and_verification — Hands-on checking of candidates through divisibility tests, factor counts, plug-ins, exhaustive tallies, and traced computations.

C24: divisibility_and_primality_checks — Testing candidate integers by divisibility rules, factor scans, and explicit primality/compositeness checks.

C32: factor_exponent_bookkeeping — Tracking prime exponents, inclusion-exclusion totals, or other ledger-style discrete counts.

C52: plug_in_value_checking — Substituting candidate values back into equations or endpoint formulas to verify they satisfy the condition.

C63: exhaustive_favorable_tally — Enumerating favorable cases or step traces one by one in a probability or algorithm-count computation.

G10: symbolic_transformation — Algebraic and trigonometric manipulation that rewrites expressions, expands formulas, isolates variables, and derives exact forms.

C3: equation_normalization — Turning a relation into a standard algebraic form by substituting, moving terms, or setting two expressions equal.

C8: line_by_line_rearrangement — Straight algebraic rewriting that carries equations forward one equality at a time.

C17: vector_constraint_equations — Writing vector, dot-product, or coordinate-condition equations that encode geometric constraints symbolically.

C19: variable_isolation_steps — Manipulations whose main role is isolating one variable or symmetric expression in terms of another.

C29: structural_relation_derivation — Deriving algebraic relations from recurrence, progression, polynomial, or symmetry assumptions.

C34: decimal_approximation_work — Converting an exact symbolic quantity into a numerical decimal estimate for comparison or reporting.

C37: closed_form_assembly — Assembling a final exact expression from previously derived pieces, radicals, or product formulas.

C60: trig_identity_expansion — Rewriting trigonometric expressions through angle-sum, product-to-sum, or equivalent identity expansions.

C61: expansion_arithmetic — Carrying out detailed polynomial, radical, or complex-number expansion arithmetic term by term.

G11: summation_and_index_identities — Compact manipulations of indexed terms such as telescoping sums/products, coefficient formulas, and series-pattern simplifications.

C6: telescoping_term_patterns — Recognizing indexed cancellation or coefficient patterns to collapse sums, products, or series formulas.