refactor: Move dual nonlinear solving to NonlinearSolveBase #330

Job	Run time
downgrade (1.10, core)	2m 21s
downgrade (1.10, downstream)	2m 29s
downgrade (1.10, wrappers)	2m 23s
test (core, 1.10, ubuntu-latest)	17m 42s
test (core, 1.10, macos-latest)	16m 23s
test (core, 1.10, windows-latest)	28m 3s
test (core, 1, ubuntu-latest)	28m 32s
test (core, 1, macos-latest)	25m 55s
test (core, 1, windows-latest)	42m 35s
test (downstream, 1.10, ubuntu-latest)	13m 5s
test (downstream, 1.10, macos-latest)	10m 7s
test (downstream, 1.10, windows-latest)	16m 6s
test (downstream, 1, ubuntu-latest)	21m 59s
test (downstream, 1, macos-latest)	15m 55s
test (downstream, 1, windows-latest)	26m 15s
test (wrappers, 1.10, ubuntu-latest)	11m 28s
test (wrappers, 1.10, macos-latest)	7m 25s
test (wrappers, 1.10, windows-latest)	15m 46s
test (wrappers, 1, ubuntu-latest)	19m 10s
test (wrappers, 1, macos-latest)	15m 14s
test (wrappers, 1, windows-latest)	23m 42s
	6h 2m 35s

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