refactor: Move dual nonlinear solving to NonlinearSolveBase #192

Job	Run time
downgrade (1.10, core)	11m 36s
downgrade (1.10, adjoint)	9m 12s
test (1.10, ubuntu-latest, core)	11m 47s
downgrade (1.10, alloc_check)	8m 53s
test (1.10, ubuntu-latest, adjoint)	9m 6s
test (1.10, ubuntu-latest, alloc_check)	9m 3s
test (1.10, macos-latest, core)	5m 52s
test (1.10, macos-latest, adjoint)	2m 40s
test (1.10, macos-latest, alloc_check)	1m 46s
test (1.10, windows-latest, core)	12m 12s
test (1.10, windows-latest, adjoint)	10m 52s
test (1.10, windows-latest, alloc_check)	10m 27s
test (1, ubuntu-latest, core)	20m 21s
test (1, ubuntu-latest, adjoint)	15m 32s
test (1, ubuntu-latest, alloc_check)	15m 7s
test (1, macos-latest, core)	1s
test (1, macos-latest, adjoint)	1s
test (1, macos-latest, alloc_check)	1s
test (1, windows-latest, core)	18m 39s
test (1, windows-latest, adjoint)	16m 40s
test (1, windows-latest, alloc_check)	16m 9s
	3h 25m 57s

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