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Equity-Based Compensation - Fair Value of Stock Option Awards (Details) - Stock Options
3 Months Ended 12 Months Ended
Dec. 31, 2021
Sep. 30, 2021
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]    
Expected volatility (percent) 55.80% 59.70%
Expected dividend yield (percent) 0.00% 0.00%
Expected term 6 years 6 years
Risk-free interest rate (percent) 1.20% 0.70%