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DIVESTITURES - Additional Information (Detail) $ in Millions |  |  | 3 Months Ended | 12 Months Ended |  |
--- | --- | --- | --- | --- | --- | ---
Nov. 02, 2020 USD ($) country | May 30, 2019 USD ($) | Jul. 03, 2021 USD ($) | Jan. 01, 2022 USD ($) | Jan. 02, 2021 USD ($) | Dec. 28, 2019 USD ($) | Dec. 31, 2021 USD ($) | Nov. 30, 2020 country
Income Statement, Balance Sheet and Additional Disclosures by Disposal Groups, Including Discontinued Operations [Line Items] |  |  |  |  |  |  |  |
Number of countries | country |  |  |  |  |  |  |  | 5
(Loss) gain on sales of businesses |  |  |  | $ (0.6) | $ (13.5) | $ 17.0 |  |
Commercial Electronic Security |  |  |  |  |  |  |  |
Income Statement, Balance Sheet and Additional Disclosures by Disposal Groups, Including Discontinued Operations [Line Items] |  |  |  |  |  |  |  |
(Loss) gain on sales of businesses |  |  |  |  | 4.2 |  |  |
Convergent Security Solutions | Discontinued Operations, Disposed of by Sale |  |  |  |  |  |  |  |
Income Statement, Balance Sheet and Additional Disclosures by Disposal Groups, Including Discontinued Operations [Line Items] |  |  |  |  |  |  |  |
Agreement for divesture of interest in consolidated subsidiaries |  |  |  |  |  |  | $ 3,200.0 |
Small Business in Security Segment |  |  |  |  |  |  |  |
Income Statement, Balance Sheet and Additional Disclosures by Disposal Groups, Including Discontinued Operations [Line Items] |  |  |  |  |  |  |  |
Number of countries | country | 5 |  |  |  |  |  |  |
Proceeds from sales of businesses, net of cash sold | $ 60.9 |  |  |  |  |  |  |
(Loss) gain on sales of businesses |  |  |  |  | (13.5) |  |  |
Small Business in Security Segment | Oil & Gas Product Line |  |  |  |  |  |  |  |
Income Statement, Balance Sheet and Additional Disclosures by Disposal Groups, Including Discontinued Operations [Line Items] |  |  |  |  |  |  |  |
(Loss) gain on sales of businesses |  |  |  |  | (17.7) |  |  |
Small Business in Security Segment | Commercial Electronic Security |  |  |  |  |  |  |  |
Income Statement, Balance Sheet and Additional Disclosures by Disposal Groups, Including Discontinued Operations [Line Items] |  |  |  |  |  |  |  |
(Loss) gain on sales of businesses |  |  | $ (1.0) |  |  |  |  |
Other Divestures |  |  |  |  |  |  |  |
Income Statement, Balance Sheet and Additional Disclosures by Disposal Groups, Including Discontinued Operations [Line Items] |  |  |  |  |  |  |  |
(Loss) gain on sales of businesses |  |  |  |  | $ 4.1 | 3.0 |  |
Sargent & Greenleaf |  |  |  |  |  |  |  |
Income Statement, Balance Sheet and Additional Disclosures by Disposal Groups, Including Discontinued Operations [Line Items] |  |  |  |  |  |  |  |
Proceeds from sales of businesses, net of cash sold |  | $ 79.0 |  |  |  |  |  |
Pre-tax gain on divestiture |  | $ 17.0 |  |  |  |  |  |
Pretax income |  |  |  |  |  | $ 4.6 |  |

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