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