
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
Among the following propositions, the correct one(s) is (are):

**(1)** The angle between the plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, $\overrightarrow{a}=(2,0)$, $|\overrightarrow{b}|=1$, then $|\overrightarrow{a}+\overrightarrow{b}|=\sqrt{7}$.

**(2)** In the triangle $\triangle ABC$, the sides opposite to $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If $a\cos C$, $b\cos B$, and $c\cos A$ form an arithmetic sequence, then $B=\frac{\pi}{3}$.

**(3)** $O$ is a fixed point on the plane of $\triangle ABC$, and the moving point $P$ satisfies: $\overrightarrow{OP}=\overrightarrow{OA}+\lambda \left( \frac{\overrightarrow{AB}}{\sin C}+ \frac{\overrightarrow{AC}}{\sin B} \right)$, where $\lambda \in (0,+\infty)$. Then the line $AP$ always passes through the incenter of $\triangle ABC$.

**(4)** Let the function $f(x) = \begin{cases} x-\lfloor x\rfloor, & x\geq 0 \\ f(x+1), & x<0 \end{cases}$ where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to x, for example, $\lfloor -1.3 \rfloor =-2$, $\lfloor 1.3 \rfloor =1$. Then, the function $y=f(x)-\frac{1}{4}x-\frac{1}{4}$ has 2 distinct non-zero roots.

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