refactor: Move dual nonlinear solving to NonlinearSolveBase #324

Job	Run time
downgrade (1.10, core)	2m 27s
downgrade (1.10, downstream)	2m 26s
downgrade (1.10, wrappers)	2m 17s
test (core, 1.10, ubuntu-latest)	19m 23s
test (core, 1.10, macos-latest)	14m 21s
test (core, 1.10, windows-latest)	30m 0s
test (core, 1, ubuntu-latest)	27m 17s
test (core, 1, macos-latest)	21m 44s
test (core, 1, windows-latest)	39m 59s
test (downstream, 1.10, ubuntu-latest)	13m 6s
test (downstream, 1.10, macos-latest)	8m 28s
test (downstream, 1.10, windows-latest)	16m 57s
test (downstream, 1, ubuntu-latest)	21m 17s
test (downstream, 1, macos-latest)	15m 7s
test (downstream, 1, windows-latest)	25m 43s
test (wrappers, 1.10, ubuntu-latest)	12m 6s
test (wrappers, 1.10, macos-latest)	8m 22s
test (wrappers, 1.10, windows-latest)	15m 49s
test (wrappers, 1, ubuntu-latest)	18m 30s
test (wrappers, 1, macos-latest)	12m 14s
test (wrappers, 1, windows-latest)	23m 21s
	5h 50m 54s

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