Given that two chords $AB$ and $CD$ intersect at point $P$ inside a circle, with $AP = 4$, $PB = 6$, $CP = 3$, and $PD = k$. If $\log(AP \cdot PB) = 1$ and $\log(CP \cdot PD) = 1$, find the value of $k$. Let's think step by step and output the final answer within \boxed{}.
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