
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 $\vec{a}=(1,3)$, $\vec{b}=(x,-1)$ and $\vec{a} \bot \vec{b}$, then $x=$ __________.

(2) In the arithmetic sequence $\{a_n\}$, if $a_1-a_4+a_8-a_{12}+a_{15}=2$, then $a_3+a_{13}=$ __________.

(3) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given vectors $\vec{m}=(a+c,b)$, $\vec{n}=(c-a,a-b)$, and $\vec{m} \bot \vec{n}$, the magnitude of angle $C$ is __________.

(4) Regarding the function $f(x)=4\sin (2x+\frac{\pi }{3} )$ (where $x\in \mathbb{R}$), the following statements are made:

- $y=f(x)$ is a periodic function with the smallest positive period of $2\pi$.
- $y=f(x)$ can be rewritten as $y=4\cos (2x-\frac{\pi }{6} )$.
- The graph of $y=f(x)$ is symmetric about the point $(-\frac{\pi }{6},0)$.
- The graph of $y=f(x)$ is symmetric about the line $x=-\frac{5\pi }{12}$.

Among these, the correct statement(s) is/are __________.

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