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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