For some integers $u$,$ v$, and $w$, the equation
$$26ab - 51bc + 74ca = 12(a^2 + b^2 + c^2)$$
holds for all real numbers a, b, and c that satisfy
$$au + bv + cw = 0$$
Find the minimum possible value of $u^2 + v^2 + w^2$.

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