Find the smallest positive integer $k$ such that 
\[(16a^2 + 36b^2 + 81c^2)(81a^2 + 36b^2 + 16c^2) < k(a^2 + b^2 + c^2)^2,\] 
for some ordered triple of positive integers $(a,b,c)$.

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