At the outset, take a certain 3-digit base ten positive integer N composed entirely of nonzero digits. The three digits may or may not be distinct.

Step-1: Add the digits of N.
Step-2: Multiply the digits of N.
Step-3: Multiply the result obtained in Step-1 by the result obtained in Step-2.
Step-4: Multiply the result obtained in Step-3 by the units digit of N.

Surprise ! The result obtained in Step-4 is N itself !!

The value of N having this property is unique. What is that value? Prove that there are no others.

*** No computer programs.