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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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