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One for the New Year

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What is the limit of the following monstrous continued fraction as N goes to infinity? Please note that this value is corrected from the original posted version. Thanks to Michael Schweitzer for pointing out the error.

```                         1  + ...
N  + -----------
1999 + ...
1999 +  ------------------
1999 + ...
1  + -----------
N  + ...
1 + -----------------------------
1999 + ...
1  +  ----------
N  + ...
N   +  ------------------
N  + ...
1999  + ----------
1  + ...
```
Notes:
• Assume the limiting value exists.
• Every fraction has one of the three forms:
```         1999 +
(a)  1 + ------
N +

1 +
(b)  N + ------         or
1999 +

N +
(c) 1999 + ------
1 +
```

Source: Angel Dorito, Crux. Math Nov 1998

HINT

Because of the error, let me spare you the trouble of starting over. It turns out that the answer is the golden ratio. The presence of 1999 is irrelevant. The limit is the golden ratio no matter what real number is used in place of 1999.

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