Let $a,$ $b,$ $c$ be positive real numbers such that $a + b + c = 1.$  Find the minimum value of $a^2 + 2b^2 + c^2.$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $a,$ $b,$ $c$ be positive real numbers such that $a + b + c = 1.$  Find the minimum value of $a^2 + 2b^2 + c^2.$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\frac{2}{5}
Let $a,$ $b,$ $c$ be positive real numbers such that $a + b + c = 1.$  Find the minimum value of $a^2 + 2b^2 + c^2.$
train