Let $a_1$, $a_2$, \ldots, $a_6$; $b_1$, $b_2$, \ldots, $b_6$; and $c_1$, $c_2$, \ldots, $c_6$ be permutations of $1$, $2$, \ldots, $6$, respectively. Find the minimum value of $\sum_{i=1}^{6} a_i b_i c_i$.

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