For which real numbers $c$ is there a straight line that intersects the curve  
$$  
y=x^{4}+9 x^{3}+c x^{2}+9 x+4  
$$  
in four distinct points? The original answer is in the format $c < \frac{m}{n}$. Please provide the value of $m + n$.

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