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Algebra Level 5

Fix the constant $$A$$. The greatest value of $$(A+X)^3 \times (A-X)^4$$ over the domain $$X \leq A$$ can be expressed as

$\frac{ p^a \times q^b \times A ^c} { r ^ d }$

for prime numbers $$p,q,$$ and $$r$$ and positive integers $$a, b, c, d$$.

Find $$p + q + r + a + b+c+d$$.

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