Hello Algebra? It's Me Calculus!

Calculus Level 4

\[\begin{cases}\displaystyle f(x) = x + \int_0^1 t (x+t) f(t) \, dt \\ g(x) = a \left( \dfrac{23}6 f(x) \right)^2 + b \left( \dfrac{23}6 f(x)\right) + c \end{cases} \]

Let \(f\) and \(g\) be two functions as defined above, where \(a\), \(b\) and \(c\) are non-zero constants and the coefficients of \(x^2\), \(x\) and \(1\) in the function \(g(x)\) are equal. Find \( \dfrac{b+c}a - 1\).

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