Convergence of the series $\sum_{n}\frac{3^{n}n!}{7 \cdot 10 \cdot 13 \cdots (3n+4)}x^{n}$, $x >0$

Question:Convergence of the series $\sum_{n}\frac{3^{n}n!}{7 \cdot 10 \cdot 13 \cdots (3n+4)}x^{n}$, $x >0$. I want to know its convergence at $x=1$.

Context:I have found that the radius of the convergence of the above series is 1 by following means:

$$1 \geq \frac{n!}{\frac{7 \cdot 10 \cdot 13 \cdots (3n+4)}{3^{n}}} \geq \frac{1}{(n+1)(n+2)}$$ and then raising every term by $\frac{1}{n}$ and then taking limit $n$ tends to $\infty$ followed by sandwich theorem. But I am not able to conclude what happens to the series at $x=1$.

The question has answer here, but it seemed to be non rigorous and didn't help much.