Geometry Level 3

For real $$x$$, define $$f(x)=\sin \left( \frac{\pi}{6} \sin \left( \frac{\pi}{2} \sin x \right) \right)$$ and $$g(x)=\frac{\pi}{2} \sin x$$. Let $$(fog)(x)$$ denotes $$f(g(x))$$ and $$(gof)(x)$$ denotes $$g(f(x))$$. Then which of the following is/are true?
$\begin{array}{} (1) \, \text{Range of } \ f \text{ is} \left[-\frac{1}{2},\frac{1}{2} \right] \quad & (2) \, \text{Range of } \ fog \text{ is} \left[-\frac{1}{2},\frac{1}{2} \right] \\ (3) \, \displaystyle \lim_{x\to 0} \frac{f(x)}{g(x)}=\frac{\pi}{6} & (4) \, \text{ There is an } x \in \mathbb{R} \ \text{such that} \ (gof)(x)=1 \end{array}$

Submit your answer as the increasing order of the serial numbers of all the correct options. For example, if your answer is $$(1),(2)$$, then submit 12 as the correct answer, if your answer is $$(2),(3),(4)$$, then submit 234 as the correct answer.

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