Smallest prime factor of $F_{F_{38}}+F_{38}+1$ wanted

Question

This question is related to this one.

Let $F_n$ be the $n$-th fibonacci number and $p(n)$ be the smallest prime factor of $$f(n):=F_{F_n}+F_n+1$$

The values $p(n)$ except for $n=11$ and $n=38$ upto $n=44$ are :

1  3
2  3
3  2
4  2
5  11
6  2
7  13
8  2
9  2
10  3
12  137
13  233
14  5
15  2
16  2
17  5
18  2
19  89
20  2
21  2
22  89
23  3
24  5
25  3
26  3
27  2
28  2
29  19
30  2
31  6553
32  2
33  2
34  3
35  11
36  43
37  5
39  2
40  2
41  941041
42  2
43  5
44  2

For $n=11$ , we get a prime number : $$p(11)=f(11)=1779979416004714279$$

What is the smallest prime factor of $f(38)$ , the remaining case. According to my search , there is no prime factor below $6\cdot 10^9$ , a doublecheck is welcome!

$f(38)$ has $8\ 168\ 944$ digits , so it is almost surely composite. To prove it , we need a prime factor , or a very long test. But I prefer a prime factor.