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Derivative Financial Instruments and Fair Value Measurements - Fair Value Measurements (Details) - USD ($)
$ in Millions
12 Months Ended
Sep. 30, 2023
Sep. 30, 2022
Interest Rate Swap and Interest Rate Cap | Other current assets    
Derivative financial instruments    
Fair values of the interest rate swap and cap agreements $ 17.2  
Interest Rate Swap and Interest Rate Cap | Other non-current assets    
Derivative financial instruments    
Fair values of the interest rate swap and cap agreements 37.5  
Interest Rate Swap | Other current assets    
Derivative financial instruments    
Fair values of the interest rate swap and cap agreements   $ 9.4
Interest Rate Swap | Other non-current assets    
Derivative financial instruments    
Fair values of the interest rate swap and cap agreements   $ 41.8
Designated as Hedging Instrument | Cash Flow Hedging | Interest Rate Swap    
Derivative financial instruments    
Losses recognized in income $ 0.0