Let $a,$ $b,$ $c$ be positive real numbers such that $a + b + c = 1.$  Find the minimum value of $a^2 + 2b^2 + c^2.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Let's think step by step and output the final answer within \boxed{}.
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