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Maximize[{Min[a, 1-2a, b, 1/2 (1-2b), c, d, 1-2c-2d, e, f, 1/2 (1-2e-2f)], e == 1/3-a-b-c && f == 2a+b-c-d-1/6}, {a, b, c, d, e, f}]