The number $145$ is well known for the property that the sum of the factorial of its digits is equal to $145$: $$1! + 4! + 5! = 1 + 24 + 120 = 145.$$

Perhaps less well known is $169$, in that it produces the longest chain of numbers that link back to $169$; it turns out that there are only three such loops that exist:

$$\begin{align} &169 \to 363601 \to 1454 \to 169\\ &871 \to 45361 \to 871\\ &872 \to 45362 \to 872 \end{align}$$

It is not difficult to prove that EVERY starting number will eventually get stuck in a loop. For example,

$$\begin{align} &69 \to 363600 \to 1454 \to 169 \to 363601 (\to 1454)\\ &78 \to 45360 \to 871 \to 45361 (\to 871)\\ &540 \to 145 (\to 145) \end{align}$$

Starting with $69$ produces a chain of five non-repeating terms, but the longest non-repeating chain with a starting number below one million is sixty terms.

How many chains, with a starting number below one million, contain exactly sixty non-repeating terms?