For a positive four-digit integer $n,$ let $T(n)$ be the number created by swapping the hundreds and thousands digits of $n$ and swapping the tens and units digits of $n.$ There is a unique integer $M$ such that $T(M)= 4M.$ What are the last three digits of $M$?

This problem is posed by Michael T.

**Details and assumptions**

For example, $T(1234)=2143$ and $T(1508) = 5180.$

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