
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
Given unit vectors $\overrightarrow{i}, \overrightarrow{j}, \overrightarrow{k}$ with the angle between any two of them being $\theta (0 < \theta < \pi$ and $\theta \neq \frac{\pi}{2})$, if a spatial vector $\overrightarrow{a}$ satisfies $\overrightarrow{a}=x \overrightarrow{i}+y \overrightarrow{j}+z \overrightarrow{k} (x,y,z \in \mathbb{R})$, then the ordered real number tuple $(x,y,z)$ is called the "affine" coordinate of vector $\overrightarrow{a}$ in the "affine" coordinate system $O-xyz$ (where $O$ is the origin) and is denoted as $\overrightarrow{a}=(x,y,z)_{\theta}$. The following propositions are given:  
(1) Given $\overrightarrow{a}=(1,3,-2)_{\theta}, \overrightarrow{b}=(4,0,2)_{\theta}$, then $\overrightarrow{a} \cdot \overrightarrow{b}=0$;  
(2) Given $\overrightarrow{a}=(x,y,0)_{\frac{\pi}{3}}, \overrightarrow{b}=(0,0,z)_{\frac{\pi}{3}}$ where $xyz \neq 0$, then the angle between vector $\overrightarrow{a}$ and $\overrightarrow{b}$ reaches its minimum value if and only if $x=y$;  
(3) Given $\overrightarrow{a}=(x_{1},y_{1},z_{1})_{\theta}, \overrightarrow{b}=(x_{2},y_{2},z_{2})_{\theta}$, then $\overrightarrow{a}+ \overrightarrow{b}=(x_{1}+x_{2},y_{1}+y_{2},z_{1}+z_{2})_{\theta}$;  
(4) Given $\overrightarrow{OA}=(1,0,0)_{\frac{\pi}{3}}, \overrightarrow{OB}=(0,1,0)_{\frac{\pi}{3}}, \overrightarrow{OC}=(0,0,1)_{\frac{\pi}{3}}$, then the surface area $S$ of the tetrahedron $O-ABC$ is $\sqrt{2}$, among which the true propositions are __________ (Write down the numbers of all true propositions).

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