Let \(a,b\) and \(c\) be positive numbers satisfying \(a+b+c=3\). If the maximum value of \(a^2 b^3 c^2\) can be expressed as \(3^p \times 2^q \times 7^r\), where \(p,q,r\) are integers, find the value of \(p+q+r\).

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