Let $a$ and $b$ be positive real numbers.  Find the minimum value of
\[a^2 + b^2 + \frac{1}{(a + b)^2}.\]
Question:
Let $a$ and $b$ be positive real numbers.  Find the minimum value of
\[a^2 + b^2 + \frac{1}{(a + b)^2}.\]
Answer:
Let's think step by step.

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\sqrt{2}
Let $a$ and $b$ be positive real numbers.  Find the minimum value of
\[a^2 + b^2 + \frac{1}{(a + b)^2}.\]
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