Digit factorial chains

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:

\[ \begin{align} &169 \rightarrow 363601 \rightarrow 1454 \rightarrow 169\\ &871 \rightarrow 45361 \rightarrow 871\\ &872 \rightarrow 45362 \rightarrow 872 \end{align} \]

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

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

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?