Differentiate the usage of "lie" (intransitive) and "lay" (transitive) in formal written English across past tenses.

• Lie (recline/exist): Lie – Lay – Lain – Lying (e.g., “The solution lies in cooperation”).
• Lay (place/put): Lay – Laid – Laid – Laying (e.g., “The envoy laid the document on the table”).