HMMT 2015 U36

A prime number $p$ is twin if at least one of $p+2$ or $p-2$ is prime and sexy if at least one of $p+6$ and $p-6$ is prime. How many sexy twin primes (i.e. primes that are both twin and sexy) are there less than $10^{9}$ ? Express your answer as a positive integer $N$ in decimal notation; for example, 521495223 . If your answer is in this form, your score for this problem will be $\max \left\{0,25-\left\lfloor\frac{1}{10000}|A-N|\right\rfloor\right\}$, where $A$ is the actual answer to this problem. Otherwise, your score will be zero.