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Acquisitions and Disposition - Acquisitions Narrative (Details) - USD ($) $ in Millions |  |  |  |  |  | 3 Months Ended | 12 Months Ended
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May 02, 2022 | Mar. 07, 2022 | Apr. 05, 2021 | Nov. 18, 2020 | Jan. 14, 2020 | Jan. 01, 2023 | Oct. 02, 2022 | Apr. 03, 2022 | Jan. 02, 2022 | Oct. 03, 2021 | Jul. 04, 2021 | Apr. 04, 2021 | Jun. 28, 2020 | Jan. 01, 2023 | Jan. 02, 2022 | Jan. 03, 2021
Business Acquisition [Line Items] |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Payment for contingent consideration liability, financing activities |  |  |  |  |  |  |  |  |  |  |  |  |  | $ 3.3 | $ 1.6 | $ 0.0
Gain on insurance settlement |  |  |  |  |  |  |  |  |  |  |  |  |  | 0.0 | 19.0 | 0.0
Goodwill impairment charge |  |  |  |  |  | $ 10.3 | $ 30.7 |  |  |  |  |  |  | 41.0 | 0.0 | 147.7
Disposal Group, Not Discontinued Operations | Russian Operations |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Business Acquisition [Line Items] |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Discontinued Operation, Income (Loss) from Discontinued Operation During Phase-out Period, before Income Tax, Attributable to Parent |  |  |  |  |  |  |  |  |  |  |  |  |  | 1.4 | 3.2 | 2.4
Softworld, Inc. |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Business Acquisition [Line Items] |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Ownership percentage acquired (percent) |  |  | 100.00% |  |  |  |  |  |  |  |  |  |  |  |  |
Purchase price of acquisition |  |  | $ 215.0 |  |  |  |  |  |  |  |  |  |  |  |  |
Purchase price paid at closing |  |  | 220.4 |  |  |  |  |  |  |  |  |  |  |  |  |
Additional consideration payable |  |  | 2.6 |  |  |  |  |  |  |  |  |  |  |  |  |
Intangible assets acquired |  |  | 79.4 |  |  |  |  |  |  |  |  |  |  |  |  |
Net working capital adjustment |  |  | (6.0) |  |  |  |  |  |  | $ (6.0) |  |  |  |  |  |
Pro forma revenues |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 4,940.9 | 4,626.5
Pro forma net earnings (loss) |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 157.7 | (70.8)
Amortization of intangible assets |  |  |  |  |  |  |  |  | $ 2.0 | 2.0 | $ 2.0 | $ 2.0 |  |  |  |
Gain on insurance settlement |  |  |  |  |  |  |  |  | $ 19.0 |  |  |  |  |  |  |
Pro forma information, revenue of acquiree since acquisition date, actual |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 98.0 |
Pro forma information, earnings or loss of acquiree since acquisition date, actual |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 4.7 |
Softworld, Inc. | Customer relationships |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Business Acquisition [Line Items] |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Intangible assets acquired |  |  | $ 54.9 |  |  |  |  |  |  |  |  |  |  |  |  |
Acquired finite-lived intangible assets, weighted average useful life (years) |  |  | 10 years |  |  |  |  |  |  |  |  |  |  |  |  |
Softworld, Inc. | Non-compete agreements |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Business Acquisition [Line Items] |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Intangible assets acquired |  |  | $ 1.4 |  |  |  |  |  |  |  |  |  |  |  |  |
Acquired finite-lived intangible assets, weighted average useful life (years) |  |  | 5 years |  |  |  |  |  |  |  |  |  |  |  |  |
Softworld, Inc. | Trade names |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Business Acquisition [Line Items] |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Intangible assets acquired |  |  | $ 23.1 |  |  |  |  |  |  |  |  |  |  |  |  |
Acquired finite-lived intangible assets, weighted average useful life (years) |  |  | 10 years |  |  |  |  |  |  |  |  |  |  |  |  |
Greenwood/Asher |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Business Acquisition [Line Items] |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Ownership percentage acquired (percent) |  |  |  | 100.00% |  |  |  |  |  |  |  |  |  |  |  |
Purchase price of acquisition |  |  |  | $ 3.5 |  |  |  |  |  |  |  |  |  |  |  |
Purchase price paid at closing |  |  |  | 5.2 |  |  |  |  |  |  |  |  |  |  |  |
Contingent consideration, liability |  |  |  | $ 2.1 |  | 3.3 |  |  |  |  |  |  |  | 3.3 |  |
Expense related to earnout liability |  |  |  |  |  |  |  |  |  |  |  |  |  | 1.0 | 2.5 |
Payment for contingent consideration liability, financing activities |  |  |  |  |  |  |  | $ 0.7 |  |  |  |  |  |  |  |
Payment for contingent consideration liability |  |  |  |  |  |  |  | $ 2.3 |  |  |  |  |  |  |  |
Insight |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Business Acquisition [Line Items] |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Ownership percentage acquired (percent) |  |  |  |  | 100.00% |  |  |  |  |  |  |  |  |  |  |
Purchase price of acquisition |  |  |  |  | $ 34.5 |  |  |  |  |  |  |  |  |  |  |
Purchase price paid at closing |  |  |  |  | 38.1 |  |  |  |  |  |  |  |  |  |  |
Contingent consideration, liability |  |  |  |  | $ 1.6 |  |  |  |  |  |  |  |  |  |  |
Expense related to earnout liability |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 0.1 | $ 0.1
Payment for contingent consideration liability, financing activities |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 1.6 |
Payment for contingent consideration liability |  |  |  |  |  |  |  |  |  | $ 1.8 |  |  |  |  |  |
Payments for previous acquisition |  |  |  |  |  |  |  |  |  |  |  |  | $ 0.1 |  |  |
Pediatric Therapeutic Services |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Business Acquisition [Line Items] |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Ownership percentage acquired (percent) | 100.00% |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Purchase price of acquisition | $ 82.1 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Purchase price paid at closing | 85.7 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Additional consideration payable | 1.1 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Employee retention credits |  |  |  |  |  |  | $ 0.1 |  |  |  |  |  |  |  |  |
Intangible assets acquired | 40.3 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Pro forma information, revenue of acquiree since acquisition date, actual |  |  |  |  |  |  |  |  |  |  |  |  |  | 28.5 |  |
Pro forma information, earnings or loss of acquiree since acquisition date, actual |  |  |  |  |  |  |  |  |  |  |  |  |  | 3.8 |  |
Pediatric Therapeutic Services | Customer relationships |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Business Acquisition [Line Items] |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Intangible assets acquired | $ 29.8 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Acquired finite-lived intangible assets, weighted average useful life (years) | 15 years |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Pediatric Therapeutic Services | Non-compete agreements |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Business Acquisition [Line Items] |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Intangible assets acquired | $ 1.2 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Acquired finite-lived intangible assets, weighted average useful life (years) | 5 years |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Pediatric Therapeutic Services | Trade names |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Business Acquisition [Line Items] |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Intangible assets acquired | $ 9.3 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Acquired finite-lived intangible assets, weighted average useful life (years) | 15 years |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
RocketPower |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Business Acquisition [Line Items] |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Ownership percentage acquired (percent) |  | 100.00% |  |  |  |  |  |  |  |  |  |  |  |  |  |
Purchase price of acquisition |  | $ 59.3 |  |  |  |  |  |  |  |  |  |  |  |  |  |
Purchase price paid at closing |  | 61.8 |  |  |  |  |  |  |  |  |  |  |  |  |  |
Additional consideration payable |  | 1.1 |  |  |  |  |  |  |  |  |  |  |  |  |  |
Contingent consideration, liability |  | 0.6 |  |  |  |  |  |  |  |  |  |  |  |  |  |
Intangible assets acquired |  | 15.8 |  |  |  |  |  |  |  |  |  |  |  |  |  |
Pro forma information, revenue of acquiree since acquisition date, actual |  |  |  |  |  |  |  |  |  |  |  |  |  | 24.3 |  |
Pro forma information, earnings or loss of acquiree since acquisition date, actual |  |  |  |  |  |  |  |  |  |  |  |  |  | (43.5) |  |
Possible future contingent consideration |  | 31.8 |  |  |  |  |  |  |  |  |  |  |  |  |  |
Goodwill expected to be deductible |  |  |  |  |  | $ 27.5 |  |  |  |  |  |  |  | 27.5 |  |
Goodwill impairment charge |  |  |  |  |  |  |  |  |  |  |  |  |  | $ 41.0 |  |
RocketPower | Customer relationships |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Business Acquisition [Line Items] |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Intangible assets acquired |  | $ 7.5 |  |  |  |  |  |  |  |  |  |  |  |  |  |
Acquired finite-lived intangible assets, weighted average useful life (years) |  | 3 years |  |  |  |  |  |  |  |  |  |  |  |  |  |
RocketPower | Non-compete agreements |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Business Acquisition [Line Items] |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Intangible assets acquired |  | $ 1.7 |  |  |  |  |  |  |  |  |  |  |  |  |  |
Acquired finite-lived intangible assets, weighted average useful life (years) |  | 6 years |  |  |  |  |  |  |  |  |  |  |  |  |  |
RocketPower | Trade names |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Business Acquisition [Line Items] |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Intangible assets acquired |  | $ 6.6 |  |  |  |  |  |  |  |  |  |  |  |  |  |
Acquired finite-lived intangible assets, weighted average useful life (years) |  | 10 years |  |  |  |  |  |  |  |  |  |  |  |  |  |
Acquisition-related Costs | Softworld, Inc. |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Business Acquisition [Line Items] |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Net Income (Loss) |  |  |  |  |  |  |  |  |  |  |  |  |  |  | $ 1.3 |

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