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Stock-Based Compensation - Assumptions Used to Value Stock Options (Details) - $ / shares
9 Months Ended
Jun. 30, 2024
Jun. 30, 2023
Share-Based Payment Arrangement [Abstract]    
Expected dividend yield 0.00% 0.00%
Risk-free interest rate   3.69%
Expected volatility   86.40%
Expected life (in years)   6 years 3 months
Weighted average grant date fair value per share of options granted (in dollars per share)   $ 24.80