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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