A number chain is created by continuously adding the square of the digits in a number to form a new number until it has been seen before.

For example, $$\begin{array}{rl}& 44\to 32\to 13\to 10\to \mathbf{1}\to \mathbf{1}\\ & 85\to \mathbf{89}\to 145\to 42\to 20\to 4\to 16\to 37\to 58\to \mathbf{89}\end{array}$$

Therefore any chain that arrives at $1$ or $89$ will become stuck in an endless loop. What is most amazing is that EVERY starting number will eventually arrive at $1$ or $89$.

How many starting numbers below ten million will arrive at $89$?