The equation $ax^3 + bx^2 + cx + d = 0$ has three distinct solutions. How many distinct solutions does the following equation have:

$4(ax^3 + bx^2 + cx + d)(3ax + b) = (3ax^2 + 2bx + c)^2$?

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