refactor: Move dual nonlinear solving to NonlinearSolveBase #184

Job	Run time
downgrade (1.10, core)	12m 1s
downgrade (1.10, adjoint)	9m 3s
downgrade (1.10, alloc_check)	9m 12s
test (1.10, ubuntu-latest, core)	11m 49s
test (1.10, ubuntu-latest, adjoint)	9m 23s
test (1.10, ubuntu-latest, alloc_check)	9m 13s
test (1.10, macos-latest, core)	7m 3s
test (1.10, macos-latest, adjoint)	6m 45s
test (1.10, macos-latest, alloc_check)	8m 3s
test (1.10, windows-latest, core)	12m 52s
test (1.10, windows-latest, adjoint)	10m 34s
test (1.10, windows-latest, alloc_check)	11m 23s
test (1, ubuntu-latest, core)	21m 56s
test (1, ubuntu-latest, adjoint)	15m 57s
test (1, ubuntu-latest, alloc_check)	15m 9s
test (1, macos-latest, core)	8m 28s
test (1, macos-latest, adjoint)	12m 39s
test (1, macos-latest, alloc_check)	13m 42s
test (1, windows-latest, core)	18m 35s
test (1, windows-latest, adjoint)	17m 40s
test (1, windows-latest, alloc_check)	16m 42s
	4h 18m 9s

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