Product puzzle

Write a single positive digit \((\leq 6)\) into each empty square in the figure below, such that every number \((1, 2, 3, 4, 5, 6)\) appears in each row and in each column. The number in a circle shows the product of the four numbers around it.

If you are finished, find the minimum value of the product of the numbers in the pink squares!

\(\)
Note: Here is a sample, with digits \(1, 2, 3, 4\):

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