refactor: Move dual nonlinear solving to NonlinearSolveBase #195

Job	Run time
downgrade (1.10, core)	12m 17s
downgrade (1.10, adjoint)	9m 6s
downgrade (1.10, alloc_check)	9m 11s
test (1.10, ubuntu-latest, core)	11m 55s
test (1.10, ubuntu-latest, adjoint)	9m 17s
test (1.10, ubuntu-latest, alloc_check)	9m 18s
test (1.10, macos-latest, core)	7m 29s
test (1.10, macos-latest, adjoint)	5m 47s
test (1.10, macos-latest, alloc_check)	8m 48s
test (1.10, windows-latest, core)	12m 14s
test (1.10, windows-latest, adjoint)	10m 36s
test (1.10, windows-latest, alloc_check)	10m 24s
test (1, ubuntu-latest, core)	19m 40s
test (1, ubuntu-latest, adjoint)	16m 31s
test (1, ubuntu-latest, alloc_check)	15m 35s
test (1, macos-latest, core)	8m 23s
test (1, macos-latest, adjoint)	10m 38s
test (1, macos-latest, alloc_check)	12m 3s
test (1, windows-latest, core)	18m 39s
test (1, windows-latest, adjoint)	17m 27s
test (1, windows-latest, alloc_check)	16m 18s
	4h 11m 36s

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