That number is so selfish

Find the smallest 3 digit narcissistic number that has a tens digit of 5.

This one is meant to be done by hand given the following hint:

This requires very little work if you guess and check in a smart way

Details:

A narcissistic number is a number that when its digits are individually raised to the n-th power and summed, they equal the original number.

Example:

\(407=4^3+0^3+7^3\)

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