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Subsequent Events (Additional Information) (Details) - USD ($) $ in Millions	Apr. 03, 2026	Nov. 07, 2025	Aug. 08, 2025
Revolving Credit Facility
Subsequent Event [Line Items]
Maximum borrowing capacity	$ 305.0	$ 260.0	$ 125.0
Subsequent Event
Subsequent Event [Line Items]
Minimum liquidity	381.3
Debt Issuance costs	1.0
Subsequent Event | Revolving Credit Facility
Subsequent Event [Line Items]
Maximum borrowing capacity	$ 305.0
Interest rate	0.25%
Revolving credit facility, aggregate principal amounts	$ 45.0
Subsequent Event | Base Rate [Member] | Revolving Credit Facility
Subsequent Event [Line Items]
Variable rate	2.25%
Subsequent Event | Secured Overnight Financing Rate (SOFR) [Member] | Revolving Credit Facility
Subsequent Event [Line Items]
Variable rate	3.25%

X

- Definition

Minimum Liquidity Maintenance Covenant

+ References

No definition available.

+ Details

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Namespace Prefix:	fly_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	instant

X

- Definition

Percentage points added to the reference rate to compute the variable rate on the debt instrument.

+ References

No definition available.

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X

- Definition

Amount, after accumulated amortization, of debt issuance costs. Includes, but is not limited to, legal, accounting, underwriting, printing, and registration costs.

+ References

Reference 1: http://www.xbrl.org/2009/role/commonPracticeRef
-Topic 835
-SubTopic 30
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-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481139/470-20-50-1D

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X

- Definition

The carrying value as of the balance sheet date of the current and noncurrent portions of long-term obligations drawn from a line of credit, which is a bank's commitment to make loans up to a specific amount. Examples of items that might be included in the application of this element may consist of letters of credit, standby letters of credit, and revolving credit arrangements, under which borrowings can be made up to a maximum amount as of any point in time conditional on satisfaction of specified terms before, as of and after the date of drawdowns on the line. Includes short-term obligations that would normally be classified as current liabilities but for which (a) postbalance sheet date issuance of a long term obligation to refinance the short term obligation on a long term basis, or (b) the enterprise has entered into a financing agreement that clearly permits the enterprise to refinance the short-term obligation on a long term basis and the following conditions are met (1) the agreement does not expire within 1 year and is not cancelable by the lender except for violation of an objectively determinable provision, (2) no violation exists at the BS date, and (3) the lender has entered into the financing agreement is expected to be financially capable of honoring the agreement.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.5-02(22))
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-Topic 944
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-Subparagraph (SX 210.7-03(a)(16)(a)(2))
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147478777/944-210-S99-1

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X

- Definition

The effective interest rate during the reporting period.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Topic 210
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
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X

- Definition

Maximum borrowing capacity under the credit facility without consideration of any current restrictions on the amount that could be borrowed or the amounts currently outstanding under the facility.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Topic 210
-SubTopic 10
-Name Accounting Standards Codification
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X

- Definition

Detail information of subsequent event by type. User is expected to use existing line items from elsewhere in the taxonomy as the primary line items for this disclosure, which is further associated with dimension and member elements pertaining to a subsequent event.

+ References

Reference 1: http://www.xbrl.org/2003/role/disclosureRef
-Topic 830
-SubTopic 30
-Name Accounting Standards Codification
-Section 50
-Paragraph 2
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147481674/830-30-50-2

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