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Stockholders' Equity - Accumulated Other Comprehensive Income (Details) - USD ($)
$ in Thousands
 12 Months Ended
Sep. 30, 2022
Sep. 30, 2021
Sep. 30, 2020
Increase (Decrease) in Accumulated Other Comprehensive Income [Roll Forward]      
Beginning Balance $ 1,325,334 $ 1,213,614 $ 1,138,954
Ending Balance 3,363,386 1,325,334 1,213,614
AOCI Attributable to Parent [Member]      
Increase (Decrease) in Accumulated Other Comprehensive Income [Roll Forward]      
Beginning Balance 19,351 21,919 3,511
Other comprehensive (loss) income before reclassifications (86,742) (2,589) 18,379
Amounts reclassified from accumulated other comprehensive income (16,525) 21 29
Ending Balance (83,916) 19,351 21,919
Currency Translation Adjustments      
Increase (Decrease) in Accumulated Other Comprehensive Income [Roll Forward]      
Beginning Balance 20,139 23,061 4,184
Other comprehensive (loss) income before reclassifications (169,266) (2,922) 18,877
Amounts reclassified from accumulated other comprehensive income (16,567)    
Ending Balance (165,694) 20,139 23,061
Unrealized Gains (Losses) on Available-for-Sale Securities      
Increase (Decrease) in Accumulated Other Comprehensive Income [Roll Forward]      
Beginning Balance (1) (1) (8)
Other comprehensive (loss) income before reclassifications (10,908)   5
Amounts reclassified from accumulated other comprehensive income     2
Ending Balance (10,909) (1) (1)
Unrealized Gain on Derivative Asset Net of Tax      
Increase (Decrease) in Accumulated Other Comprehensive Income [Roll Forward]      
Other comprehensive (loss) income before reclassifications 93,020    
Ending Balance 93,020    
Pension Liability Adjustments      
Increase (Decrease) in Accumulated Other Comprehensive Income [Roll Forward]      
Beginning Balance (787) (1,141) (665)
Other comprehensive (loss) income before reclassifications 412 333 (503)
Amounts reclassified from accumulated other comprehensive income 42 21 27
Ending Balance $ (333) $ (787) $ (1,141)