Triangle Complex Strike Again

Algebra Level pending

For some positive real number \(c,\) the three complex roots of the equation \(x^3+cx+1=0\) form a triangle of area \(\frac{3}{25}\sqrt{109}.\) If \(c\) can be expressed as \(\frac{a}{b}\) for coprime positive integers \(a\) and \(b,\) what is \(a+b?\)

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