Suppose \(a\) and \(b\) are positive integers satisfying \( 1 \leq a \leq 31\), \(1 \leq b \leq 31\) such that the polynomial \(P(x)=x^3-ax^2+a^2b^3x+9a^2b^2\) has roots \(r\), \(s\), and \(t\).

Given that there exists a positive integer \(k\) such that \((r+s)(s+t)(r+t)=k^2\), compute the maximum possible value of \(ab\).

This problem is posed by Samir K.

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