Stock Option Plans - Liberty Stock Option Plan (Details) - Liberty 2004 Stock Option Plan - Stock options $ / shares in Units, $ in Thousands	1 Months Ended	9 Months Ended	12 Months Ended
Feb. 28, 2018 $ / shares shares	Sep. 30, 2022 USD ($) $ / shares shares	Dec. 31, 2021 $ / shares shares
Equity Compensation
Contractual life	10 years
Vesting period (in years)	5 years
Conversion ratio, stock based compensation	0.2857
Number of options converted	48,225
Exercise price of option converted \| $ / shares	$ 1.18
Number of options outstanding after conversion	13,771
Shares
Outstanding at the beginning of year		5,761
Exercised		(1,028)
Outstanding at the end of year		4,733	5,761
Options exercisable at end of year		4,733
Weighted Average Exercise Price
Balance at the beginning of year \| $ / shares		$ 4.14
Exercised \| $ / shares		4.14
Balance at the end of year \| $ / shares	$ 4.14	4.14	$ 4.14
Options exercisable at end of year \| $ / shares		$ 4.14
Weighted-Average Remaining Contractual Life
Outstanding at beginning of year		5 months 23 days	1 year 2 months 23 days
Outstanding at end of year		5 months 23 days	1 year 2 months 23 days
Intrinsic Value \| $		$ 22,482

The number of options issued in conversion in a noncash transaction.

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The number of options converted in a noncash transaction in a business combination.

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Weighted average price of the options converted in a business combination.

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Ratio applied to the conversion of options in a business combination.

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Period over which grantee's right to exercise award under share-based payment arrangement is no longer contingent on satisfaction of service or performance condition, in 'PnYnMnDTnHnMnS' format, for example, 'P1Y5M13D' represents reported fact of one year, five months, and thirteen days. Includes, but is not limited to, combination of market, performance or service condition.

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Line items represent financial concepts included in a table. These concepts are used to disclose reportable information associated with domain members defined in one or many axes to the table.

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The number of shares into which fully or partially vested stock options outstanding as of the balance sheet date can be currently converted under the option plan.

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The weighted-average price as of the balance sheet date at which grantees can acquire the shares reserved for issuance on vested portions of options outstanding and currently exercisable under the stock option plan.

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Amount by which the current fair value of the underlying stock exceeds the exercise price of options outstanding.

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Number of options outstanding, including both vested and non-vested options.

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A roll forward is a reconciliation of a concept from the beginning of a period to the end of a period.

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Weighted average price at which grantees can acquire the shares reserved for issuance under the stock option plan.

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Weighted average price at which option holders acquired shares when converting their stock options into shares.

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Period from grant date that an equity-based award expires, in 'PnYnMnDTnHnMnS' format, for example, 'P1Y5M13D' represents the reported fact of one year, five months, and thirteen days.

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Weighted average remaining contractual term for option awards outstanding, in 'PnYnMnDTnHnMnS' format, for example, 'P1Y5M13D' represents the reported fact of one year, five months, and thirteen days.

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Number of share options (or share units) exercised during the current period.

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