Her name is Cheryl.

I have an integer (in decimal representation) such that if I reverse its digits and add them up, I will get a new integer. I repeat this process until the resulting integer is a palindrome. We will denote an integer as a near-symmetric number if after twenty-five iterations, the resulting integer is still not a palindrome. What is the smallest positive near-symmetric number?

Details and assumptions

• A palindrome is a number that remains the same when its digits are reversed.

• As an explicit example, consider the integer to be $49$. The resulting number will be $49 + 94 = 143$. Repeat: $143 + 341 = 484$, which is a palindrome (after $2 < 25$ iterations). Thus $49$ is not a near-symmetric number.