refactor: Move dual nonlinear solving to NonlinearSolveBase #319

Job	Run time
downgrade (1.10, core)	2m 24s
downgrade (1.10, downstream)	2m 25s
downgrade (1.10, wrappers)	2m 31s
test (core, 1.10, ubuntu-latest)	15m 17s
test (core, 1.10, macos-latest)	13m 21s
test (core, 1.10, windows-latest)	24m 22s
test (core, 1, ubuntu-latest)	25m 5s
test (core, 1, macos-latest)	17m 25s
test (core, 1, windows-latest)	34m 58s
test (downstream, 1.10, ubuntu-latest)	13m 36s
test (downstream, 1.10, macos-latest)	9m 45s
test (downstream, 1.10, windows-latest)	17m 9s
test (downstream, 1, ubuntu-latest)	22m 50s
test (downstream, 1, macos-latest)	19m 13s
test (downstream, 1, windows-latest)	26m 44s
test (wrappers, 1.10, ubuntu-latest)	12m 8s
test (wrappers, 1.10, macos-latest)	7m 1s
test (wrappers, 1.10, windows-latest)	15m 52s
test (wrappers, 1, ubuntu-latest)	20m 2s
test (wrappers, 1, macos-latest)	11m 16s
test (wrappers, 1, windows-latest)	24m 1s
	5h 37m 25s

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