Algebra Level 5

Consider the following polynomial equation:$(x^2+a)^2+a=x$Where $$a$$ is a real number such that the equation has four distinct real roots. If the difference between the largest and the smallest roots of equation is $$7$$, then $$a$$ can be written in the simplest form as:$-\frac{\overline{ABBB}}{\overline{ACC}}$Find the value of $$A^2+B^2+C^2$$.

Details and Assumptions:

• $$A$$, $$B$$ and $$C$$ are some digits from $$1$$ to $$9$$.

• As an explicit example, let's say that $$-2$$, $$-1$$, $$0$$ and $$1$$ are the roots of polynomial, then the largest root is $$1$$ and the smallest is $$-2$$ and the difference between them is $$1-(-2)=3$$.

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