
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
(1) Given that the vertex of angle $θ$ coincides with the origin, its initial side coincides with the positive half-axis of $x$, and its terminal side lies on the ray $y=3x (x \geqslant 0)$, then $\sin (2θ)=$\_\_\_\_\_\_\_\_.

(2) Given $f(x)$ is an even function, when $x \leqslant 0$, $f(x)=e^{-x-1}-x$, then the equation of the tangent line to the curve $y=f(x)$ at the point $(1,2)$ is \_\_\_\_\_\_\_\_.

(3) Given the function $f(x)$ is defined as follows:
$$
f(x)=\begin{cases} 
1+\sin \pi x, & x\in (-3,0] \\
2-f(-x), & x\in (0,3] \\
\end{cases}
$$
Then, the sum of all zeros of the function $g(x)=f(x)-kx-1$ is \_\_\_\_\_\_\_\_.

(4) In $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $c=2\sqrt{3}$, $(2a-\sqrt{3}c)\cos B=\sqrt{3}b\cos C$, then the minimum value of $\frac{a}{2}+b$ is \_\_\_\_\_\_\_\_.

Present the answer in LaTex format: \boxed{Your answer}
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