[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"term-betting-kelly-criterion-de":3,"related-kelly-criterion-de":49,"mdc--ibov0j-key":65},{"id":4,"slug":5,"status":6,"section":7,"category":8,"difficulty":9,"aliases":10,"related_terms":16,"related_calculators":22,"term":25,"definition":26,"content":27,"example":28,"faq":29,"availableLocales":42},"10d4dace-0c02-4025-80ab-49d8488a472b","kelly-criterion","published","betting","strategies","advanced",[11,12,13,14,15],"Kelly formula","Kelly strategy","Kelly staking","Kelly bet","optimal f",[17,18,19,20,21],"bankroll","expected-value","edge","stake","bankroll-management",[23,24,24],"\u002Fbetting\u002Fkelly-calculator","\u002Fbetting\u002Fbankroll-growth-calculator","Kelly-Kriterium","Das Kelly-Kriterium ist eine mathematische Formel, die den optimalen Prozentsatz Ihrer Bankroll berechnet, den Sie auf eine Wette setzen sollten, basierend auf Ihrem Vorteil und den angebotenen Quoten. Es wurde 1956 von John Kelly bei Bell Labs entwickelt und maximiert das langfristige Bankroll-Wachstum bei gleichzeitiger Minimierung des Ruinrisikos. Professionelle Wettende verwenden fraktioniertes Kelly (25-50%), um die Volatilität zu reduzieren.","# Kelly-Kriterium\n\nDas **Kelly-Kriterium** ist die mathematisch optimale Formel für die Bestimmung der Einsatzhöhe. Es berechnet exakt, welchen Prozentsatz Ihrer Bankroll Sie basierend auf Ihrem Vorteil und den Quoten setzen sollten. Im Gegensatz zu Flat Staking oder willkürlichen Einheits-Systemen maximiert Kelly die geometrische Wachstumsrate Ihrer Bankroll im Laufe der Zeit. Es ist der Goldstandard für professionelle Wettende, Investoren und alle, die Risiken mit positiven Erwartungswert-Opportunitäten managen.\n\n## Inhaltsverzeichnis\n\n- [Das Kelly-Kriterium verstehen](#verstehen)\n- [Die Kelly-Formel](#formel)\n- [Schritt-für-Schritt-Berechnung](#berechnung)\n- [Fraktioniertes Kelly](#fraktioniertes)\n- [Kelly für Mehrfachwetten](#mehrfachwetten)\n- [Häufige Fehler](#fehler)\n\n## Das Kelly-Kriterium verstehen {#verstehen}\n\nStellen Sie sich vor, Sie haben eine Münze, die in 60% der Fälle Kopf zeigt, und jemand bietet Ihnen eine Wette zum gleichen Einsatz (Quote 2.0) auf Kopf an. Sie haben einen Vorteil – aber wie viel sollten Sie setzen?\n\n- Zu wenig setzen: Sie schöpfen Ihren Vorteil nicht aus.\n- Zu viel setzen: Ein schlechter Lauf löscht Ihre Bankroll aus.\n- Optimal setzen (Kelly): Maximales langfristiges Wachstum.\n\n**Das Kelly-Kriterium beantwortet:** Wie hoch ist die optimale Einsatzhöhe, um Ihren langfristigen Gewinn zu maximieren, wenn man Ihren Vorteil und die Quote berücksichtigt?\n\n**Wesentliche Erkenntnis:** Kelly maximiert nicht den erwarteten Gewinn – es maximiert den erwarteten logarithmischen Nutzen, was sich in einer maximalen geometrischen Wachstumsrate niederschlägt. Diese Unterscheidung ist entscheidend für den langfristigen Vermögensaufbau.\n\n### Warum Kelly funktioniert\n\n| Strategie | Kurzfristig | Langfristig |\n|----------|------------|-----------|\n| Alles setzen | Hohe Varianz | Bankrott |\n| Flache 1%-Einsätze | Niedriges Wachstum | Langsame Akkumulation |\n| Kelly optimal | Ausgewogen | Maximales Wachstum |\n\nKelly findet das perfekte Gleichgewicht zwischen Wachstum und Überleben. Das Verständnis Ihres [Ruinrisikos](\u002Fblog\u002Frisk-of-ruin-calculator) hilft Ihnen, die optimale Kelly-Fraktion zu bestimmen.\n\n## Die Kelly-Formel {#formel}\n\n### Grundlegende Kelly-Formel\n\n```math\nf^* = \\frac{bp - q}{b}\n```\n\nWobei:\n- f* = Anteil der Bankroll, der gesetzt werden soll\n- b = Dezimalquote - 1 (Nettoquote)\n- p = Wahrscheinlichkeit des Gewinnens\n- q = Wahrscheinlichkeit des Verlierens (1 - p)\n\n### Vereinfachte Formel\n\nFür Dezimalquoten:\n\n```math\n\\text{Kelly \\%} = \\frac{p \\times \\text{Quote} - 1}{\\text{Quote} - 1}\n```\n\nOder noch einfacher:\n\n```math\n\\text{Kelly \\%} = \\frac{\\text{Vorteil \\%}}{\\text{Quote} - 1}\n```\n\nWobei Vorteil % = (Wahrscheinlichkeit × Quote) - 1\n\n### Ableitung der Kelly-Formel\n\nDie Formel maximiert den erwarteten Log-Vermögenswert:\n\n```math\nG = p \\times \\log(1 + f \\times b) + q \\times \\log(1 - f)\n```\n\nDie Ableitung nach f und das Nullsetzen ergibt die Kelly-Formel.\n\n## Schritt-für-Schritt-Berechnung {#berechnung}\n\n### Beispiel 1: Fußballspiel\n\n**Szenario:** Sie schätzen, dass Liverpool eine 55%ige Chance hat, Chelsea zu schlagen. Bwin bietet eine Quote von 2.10 an.\n\n**Schritt 1:** Variablen identifizieren\n- Quote = 2.10\n- p (Ihre Wahrscheinlichkeit) = 0.55\n- q = 1 - 0.55 = 0.45\n- b = 2.10 - 1 = 1.10\n\n**Schritt 2:** Vorteil berechnen\n```math\n\\text{Vorteil} = (0.55 \\times 2.10) - 1 = 1.155 - 1 = 0.155 = 15.5\\%\n```\n\n**Schritt 3:** Kelly-Formel anwenden\n```math\nf^* = \\frac{0.55 \\times 2.10 - 1}{2.10 - 1} = \\frac{0.155}{1.10} = 0.141 = 14.1\\%\n```\n\n**Ergebnis:** Setzen Sie 14.1% Ihrer Bankroll.\n\n### Beispiel 2: Tennisspiel (Außenseiter)\n\n**Szenario:** Sie schätzen, dass der Außenseiter eine 35%ige Chance hat. Die Quote beträgt 3.50.\n\n- Vorteil = (0.35 × 3.50) - 1 = 0.225 = 22.5%\n- Kelly = 0.225 \u002F (3.50 - 1) = 0.225 \u002F 2.50 = **9%**\n\n### Beispiel 3: Kein Vorteil (Negatives Kelly)\n\n**Szenario:** Wahre Wahrscheinlichkeit 45%, Quote 2.00.\n\n- Vorteil = (0.45 × 2.00) - 1 = -0.10 = -10%\n- Kelly = -0.10 \u002F 1.00 = **-10%**\n\n**Negatives Kelly bedeutet, nicht zu wetten.** Wenn Sie gegen dieses Ergebnis wetten könnten, würden Sie es tun.\n\n### Kelly-Berechnungstabelle\n\n| Wahre Wahrscheinlichkeit | Quote | Vorteil | Kelly % |\n|-----------|------|------|---------|\n| 50% | 2.00 | 0% | 0% |\n| 50% | 2.20 | 10% | 8.3% |\n| 55% | 2.00 | 10% | 10% |\n| 55% | 1.90 | 4.5% | 5% |\n| 60% | 1.80 | 8% | 10% |\n| 40% | 3.00 | 20% | 10% |\n| 30% | 4.00 | 20% | 6.7% |\n\n## Fraktioniertes Kelly {#fraktioniertes}\n\n### Warum fraktioniertes Kelly verwenden?\n\nVolles Kelly setzt voraus, dass Sie Ihren genauen Vorteil kennen. In Wirklichkeit:\n- Ihre Wahrscheinlichkeitsschätzungen sind fehlerhaft\n- Stichprobengrößen sind begrenzt\n- Der Vorteil kann sich im Laufe der Zeit ändern\n\n**Fraktioniertes Kelly** (Wetten eines Bruchteils des vollen Kelly) behebt diese Probleme.\n\n### Fraktioniertes Kelly – Performance\n\n```math\n\\text{Fraktioniertes Kelly Wachstum} = f \\times G_{Kelly} \\times (2 - f)\n```\n\n| Fraktion | Wachstum vs. volles Kelly | Varianzreduktion |\n|----------|---------------------|-------------------|\n| 100% (Voll) | 100% | Keine |\n| 75% | 93.75% | Signifikant |\n| 50% (Hälfte) | 75% | Deutlich |\n| 25% (Viertel) | 43.75% | Sehr groß |\n\n**Halbes Kelly erzielt 75% des optimalen Wachstums mit weitaus geringerem Risiko.**\n\n### Empfohlene Fraktionen nach Vertrauen\n\n| Ihr Vertrauen in Ihren Vorteil | Empfohlene Fraktion |\n|---------------------|---------------------|\n| Sehr hoch (verifiziertes Modell, 1000+ Wetten) | 50-75% |\n| Hoch (gutes Modell, 500+ Wetten) | 33-50% |\n| Mittel (vernünftige Schätzungen) | 25-33% |\n| Niedrig (unsicher) | 10-25% |\n\n### Fraktioniertes Kelly – Beispiel\n\nVolles Kelly sagt, setzen Sie 14.1%. Mit halbem Kelly:\n\n```math\n\\text{Tatsächlicher Einsatz} = 14.1\\% \\times 0.50 = 7.05\\%\n```\n\n## Kelly für Mehrfachwetten {#mehrfachwetten}\n\n### Gleichzeitige, unabhängige Wetten\n\nWenn Sie mehrere Wetten gleichzeitig platzieren, reduzieren Sie die Kelly-Fraktion jeder Wette:\n\n```math\n\\text{Angepasstes Kelly}_i = \\frac{f_i^*}{\\sum f_j^*} \\times k\n```\n\nWobei k Ihre angestrebte Gesamtbelastung ist (oft auf 30-50% der Bankroll begrenzt).\n\n### Praktischer Ansatz: Diversifizierung\n\n| Anzahl der Wetten | Maximaler Einzel-Einsatz | Maximale Gesamtbelastung |\n|----------------|----------------|-------------------|\n| 1 | Volles Kelly | Volles Kelly |\n| 2-3 | 2\u002F3 Kelly jeweils | 50% gesamt |\n| 4-5 | 1\u002F2 Kelly jeweils | 50% gesamt |\n| 6+ | 1\u002F3 Kelly jeweils | 50% gesamt |\n\n### Sequentiell vs. Gleichzeitig\n\n**Sequentielle Wetten** (eine nach der anderen wird abgerechnet): Verwenden Sie jedes Mal das volle berechnete Kelly – Ihre Bankroll wird aktualisiert.\n\n**Gleichzeitige Wetten** (alle gleichzeitig ausstehend): Reduzieren Sie die Allokation, um eine Überbelastung zu verhindern.\n\n## Häufige Fehler {#fehler}\n\n### Fehler 1: Überschätzung Ihres Vorteils\n\nDer gefährlichste Fehler. Wenn Sie denken, Ihr Vorteil beträgt 10%, aber er beträgt tatsächlich 2%, wird das volle Kelly Ihre Bankroll vernichten.\n\n**Lösung:**\n- Verfolgen Sie Ihre tatsächlichen Ergebnisse über 500+ Wetten\n- Vergleichen Sie mit dem Closing Line Value (CLV)\n- Verwenden Sie fraktioniertes Kelly\n\n### Fehler 2: Ignorieren der Korrelation\n\nKelly setzt unabhängige Wetten voraus. Das Wetten auf verwandte Ergebnisse (gleiches Spiel, gleiches Team) verstößt gegen diese Annahme.\n\n**Beispiel für korrelierte Wetten:**\n- Liverpool gewinnt\n- Liverpool über 1.5 Tore\n- Liverpool ohne Gegentor\n\nDiese sind nicht unabhängig – behandeln Sie sie als eine große Wette.\n\n### Fehler 3: Bankroll nicht neu berechnen\n\nKelly-Prozentsätze sollten auf Ihre **aktuelle** Bankroll angewendet werden, nicht auf die Startbankroll.\n\n| Bankroll | 10% Kelly-Wette |\n|----------|---------------|\n| 1.000 € (Start) | 100 € |\n| 1.200 € (nach Gewinnen) | 120 € |\n| 800 € (nach Verlusten) | 80 € |\n\nDiese automatische Anpassung ist Teil der Stärke von Kelly – Sie setzen mehr, wenn Sie gewinnen, und weniger, wenn Sie verlieren.\n\n### Fehler 4: Anwendung auf negativen Erwartungswert\n\nKelly empfiehlt niemals, auf einen negativen Erwartungswert zu wetten. Wenn Ihre Berechnung eine negative Zahl oder Null ergibt, **wetten Sie nicht**.\n\n### Fehler 5: Ignorieren praktischer Einschränkungen\n\nKelly könnte vorschlagen, 25% der Bankroll zu setzen. Praktische Probleme:\n- Wettlimits des Buchmachers (z.B. bei Tipico oder Bet365)\n- Liquidität (können Sie die Wette tatsächlich platzieren?)\n- Kontosicherung\n\n## Kelly-Kriterium in der Praxis {#practice}\n\n### Der professionelle Ansatz\n\n1. **Berechnen Sie das theoretische Kelly**\n2. **Wenden Sie einen Unsicherheitsabschlag an** (typischerweise 25-50% des Kelly)\n3. **Begrenzen Sie den maximalen Einsatz** (normalerweise 5-10% unabhängig von Kelly)\n4. **Berücksichtigen Sie die Korrelation** mit anderen Wetten\n5. **Verfolgen und passen Sie an** basierend auf den Ergebnissen\n\n### Kelly mit Bankroll-Einschränkungen\n\n| Einschränkung | Anpassung |\n|------------|------------|\n| Begrenzte Bankroll | Verwenden Sie absolute Minima (10 € unabhängig von %) |\n| Wettlimits des Buchmachers | Möglicherweise müssen Sie auf verschiedene Buchmacher verteilen |\n| Konto-Langlebigkeit | Manchmal bewahrt Underbetting den Zugang |\n\n### Beispiel für ein Kelly-Wettprotokoll\n\n| Datum | Wette | Wahrscheinlichkeit | Quote | Vorteil | Volles Kelly | Tatsächlicher Einsatz | Ergebnis |\n|------|-----|------|------|------|------------|------------|--------|\n| 1. Jan | Liverpool | 55% | 2.10 | 15.5% | 14.1% | 7% | Gewinn |\n| 2. Jan | Man City | 70% | 1.50 | 5% | 10% | 5% | Gewinn |\n| 3. Jan | Chelsea | 45% | 2.40 | 8% | 5.7% | 3% | Verlust |\n\n## Kelly-Kriterium visualisiert {#visualization}\n\n### Wachstumsrate nach Einsatzhöhe\n\nFür eine Wette mit 10% Vorteil bei einer Quote von 2.0 (Kelly = 10%):\n\n| Einsatzhöhe | Erwartete Wachstumsrate |\n|----------|---------------------|\n| 0% (keine Wette) | 0% |\n| 5% | ~0.45% pro Wette |\n| 10% (Kelly) | ~0.50% pro Wette |\n| 15% | ~0.45% pro Wette |\n| 20% (2x Kelly) | 0% pro Wette |\n| 25%+ | Negatives Wachstum |\n\n**Bei 2x Kelly sinkt das erwartete Wachstum auf Null.** Darüber hinaus wird mathematisch erwartet, dass Sie trotz eines Vorteils Geld verlieren.\n\n### Ruinrisiko nach Strategie\n\n| Strategie | Risiko eines 50%igen Drawdowns |\n|----------|---------------------|\n| Volles Kelly | ~50% irgendwann |\n| Halbes Kelly | ~11% |\n| Viertel Kelly | ~1% |\n\n## Mehr erfahren\n\nFür eine umfassende Anleitung mit Beispielen aus der Praxis, Simulationsergebnissen und Tabellenvorlagen lesen Sie unseren [vollständigen Kelly-Kriterium-Leitfaden](\u002Fblog\u002Fkelly-criterion-explained).\n\n## Zugehörige Rechner\n\nWenden Sie das Kelly-Kriterium mit diesen Tools an:\n\n- [Kelly Criterion Calculator](\u002Fbetting\u002Fkelly-calculator) - Berechnen Sie die optimale Einsatzhöhe\n- [Bankroll Management](\u002Fbetting\u002Fbankroll-growth-calculator) - Planen Sie Ihre Bankroll-Strategie\n- [Bankroll Growth Calculator](\u002Fbetting\u002Fbankroll-growth-calculator) - Projizieren Sie das Wachstum im Laufe der Zeit",null,[30,33,36,39],{"answer":31,"question":32},"Wenn Sie mehr setzen als Kelly ('Overbetting'), erhöht sich Ihr Ruinrisiko drastisch, ohne dass die Rendite proportional steigt. Bei 2x Kelly sinkt Ihr erwartetes Wachstum auf Null. Oberhalb von 2x Kelly wird mathematisch erwartet, dass Sie unabhängig von Ihrem Vorteil bankrott gehen.","Was passiert, wenn ich mehr setze, als Kelly vorschlägt?",{"answer":34,"question":35},"Volles Kelly setzt voraus, dass Sie Ihren genauen Vorteil kennen, was Sie nie tun. Fraktioniertes Kelly (typischerweise 25-50%) berücksichtigt die Unsicherheit in Ihren Vorteilsschätzungen, reduziert die Varianz und bietet nahezu das gleiche langfristige Wachstum mit viel weniger Risiko. Ein 50%-Kelly-Wettender erzielt 75% des vollen Kelly-Wachstums mit weitaus geringeren Drawdowns.","Warum verwenden professionelle Wettende fraktioniertes Kelly?",{"answer":37,"question":38},"Technisch ja, aber da Casinospiele einen negativen Erwartungswert (Hausvorteil) haben, würde Kelly empfehlen, 0 € zu setzen. Kelly funktioniert nur, wenn Sie einen positiven Vorteil haben. Die einzige Casino-Ausnahme ist das Kartenzählen beim Blackjack, wo geschickte Spieler einen positiven Erwartungswert erzielen können.","Kann das Kelly-Kriterium für Casinospiele verwendet werden?",{"answer":40,"question":41},"Kelly % = Vorteil \u002F Quote. Genauer gesagt: Kelly % = (Wahrscheinlichkeit × Quote - 1) \u002F (Quote - 1). Es sagt Ihnen, mehr zu setzen, wenn Ihr Vorteil größer ist, und weniger, wenn die Quoten länger sind, um Wachstum gegen Risiko abzuwägen.","Was ist die Kelly-Kriterium-Formel in einfachen Worten?",[43,44,45,46,47,48],"pt","en","ru","de","es","tr",[50,54,58,62],{"slug":17,"section":7,"category":8,"difficulty":51,"term":52,"definition":53},"beginner","Bankroll","Eine Bankroll ist die Gesamtsumme an Geld, die ein Wetter ausschließlich für Wetten reserviert hat, vollständig getrennt von seinen persönlichen Finanzen. Richtiges Bankroll Management bestimmt die Höhe der Einsätze, schützt vor Varianz und gilt als der wichtigste Faktor für langfristigen Wetterfolg. Ohne diszipliniertes Bankroll Management werden selbst erfahrene Wetter mit positivem Erwartungswert irgendwann bankrottgehen.",{"slug":21,"section":7,"category":8,"difficulty":55,"term":56,"definition":57},"intermediate","Bankroll-Management","Bankroll Management ist der systematische Ansatz zur Bestimmung der Einsatzhöhe und zum Schutz des Wettkapitals, um Varianz zu überstehen und langfristiges Wachstum zu maximieren. Es legt fest, wie viel bei jeder Wette eingesetzt werden soll, basierend auf Vorteil, Quoten und Risikobereitschaft. Ohne angemessenes Bankroll Management droht selbst profitablen Wettern der Ruin – eine Verluststrähne von 10 Wetten mit 10% Einsatz zerstört 65% des Kapitals. Eine angemessene Einsatzstrategie sichert das Überleben in unvermeidlichen Abwärtsphasen.",{"slug":20,"section":7,"category":59,"difficulty":51,"term":60,"definition":61},"fundamentals","Einsatz","Ein Einsatz ist der Geldbetrag, der auf eine einzelne Wette gesetzt wird – dein finanzielles Risiko bei diesem spezifischen Ergebnis. Die Einsatzhöhe ist die Grundlage des Bankroll-Managements und bestimmt, wie viel du pro Wette im Verhältnis zu deinem gesamten Wettguthaben riskierst. Professionelle Wetter setzen typischerweise 1-5 % ihrer Bankroll pro Wette ein, wobei der genaue Prozentsatz durch ihren Vorteil und ihr Vertrauensniveau bestimmt wird.",{"slug":18,"section":7,"category":8,"difficulty":55,"term":63,"definition":64},"Erwartungswert (EV)","Der durchschnittliche Gewinn oder Verlust, den du langfristig von einer Wette erwarten kannst, berechnet durch Multiplikation des Werts jedes Ergebnisses mit seiner Wahrscheinlichkeit — die wichtigste Zahl, die erfolgreiche Wetter von allen anderen unterscheidet.",{"data":66,"body":67},{},{"type":68,"children":69},"root",[70,78,91,97,157,163,168,186,196,206,213,299,312,318,324,630,635,658,664,669,925,930,1141,1146,1152,1157,1523,1528,1534,1540,1550,1560,1583,1593,1913,1923,2411,2421,2427,2436,2454,2460,2469,2487,2497,2503,2685,2691,2697,2702,2720,2729,2735,3051,3149,3157,3163,3237,3243,3248,3485,3491,3497,3502,3978,3983,3989,4084,4090,4100,4110,4116,4122,4127,4135,4153,4159,4164,4172,4190,4195,4201,4213,4273,4278,4284,4296,4302,4307,4325,4331,4337,4389,4395,4456,4462,4631,4637,4643,4648,4745,4755,4761,4820,4826,4838,4844,4849],{"type":71,"tag":72,"props":73,"children":75},"element","h2",{"id":74},"kelly-kriterium",[76],{"type":77,"value":25},"text",{"type":71,"tag":79,"props":80,"children":81},"p",{},[82,84,89],{"type":77,"value":83},"Das ",{"type":71,"tag":85,"props":86,"children":87},"strong",{},[88],{"type":77,"value":25},{"type":77,"value":90}," ist die mathematisch optimale Formel für die Bestimmung der Einsatzhöhe. Es berechnet exakt, welchen Prozentsatz Ihrer Bankroll Sie basierend auf Ihrem Vorteil und den Quoten setzen sollten. Im Gegensatz zu Flat Staking oder willkürlichen Einheits-Systemen maximiert Kelly die geometrische Wachstumsrate Ihrer Bankroll im Laufe der Zeit. 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Sie haben einen Vorteil – aber wie viel sollten Sie setzen?",{"type":71,"tag":98,"props":169,"children":170},{},[171,176,181],{"type":71,"tag":102,"props":172,"children":173},{},[174],{"type":77,"value":175},"Zu wenig setzen: Sie schöpfen Ihren Vorteil nicht aus.",{"type":71,"tag":102,"props":177,"children":178},{},[179],{"type":77,"value":180},"Zu viel setzen: Ein schlechter Lauf löscht Ihre Bankroll aus.",{"type":71,"tag":102,"props":182,"children":183},{},[184],{"type":77,"value":185},"Optimal setzen (Kelly): Maximales langfristiges Wachstum.",{"type":71,"tag":79,"props":187,"children":188},{},[189,194],{"type":71,"tag":85,"props":190,"children":191},{},[192],{"type":77,"value":193},"Das Kelly-Kriterium beantwortet:",{"type":77,"value":195}," Wie hoch ist die optimale Einsatzhöhe, um Ihren langfristigen Gewinn zu maximieren, wenn man Ihren Vorteil und die Quote berücksichtigt?",{"type":71,"tag":79,"props":197,"children":198},{},[199,204],{"type":71,"tag":85,"props":200,"children":201},{},[202],{"type":77,"value":203},"Wesentliche Erkenntnis:",{"type":77,"value":205}," Kelly maximiert nicht den erwarteten Gewinn – es maximiert den erwarteten logarithmischen Nutzen, was sich in einer maximalen geometrischen Wachstumsrate niederschlägt. 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