Michael's divisible grids

There are \(N\) ways to fill each square of a \( 2 \times 3 \) rectangular grid with positive integers from 1 to 12, such that the sum of integers in each row is a multiple of 2 and the sum of integers in each column is a multiple of 3. What are the last three digits of \(N\)?

This problem is posed by Michael T.

Details and assumptions

The integers need not be distinct. As an explicit example,

\[ \begin{array}{ | l | l | l | } \hline 12 &12 &12 \\ \hline 12 & 12 & 12\\ \hline \end{array} \]

is a valid solution.

A \( 2 \times 3 \) grid has 2 rows and 3 columns.

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