Let $0 \le a,$ $b,$ $c \le 1.$  Find the maximum value of
\[\sqrt{abc} + \sqrt{(1 - a)(1 - b)(1 - c)}.\]
Question:
Let $0 \le a,$ $b,$ $c \le 1.$  Find the maximum value of
\[\sqrt{abc} + \sqrt{(1 - a)(1 - b)(1 - c)}.\]
Answer:
Let's think step by step.

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Let $0 \le a,$ $b,$ $c \le 1.$  Find the maximum value of
\[\sqrt{abc} + \sqrt{(1 - a)(1 - b)(1 - c)}.\]
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