
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) Use the Euclidean algorithm to find the greatest common divisor of $459$ and $357$ is ______.

(2) Given the function $f(x)=a\sin (\pi x+\alpha)+b\cos (\pi x+\beta)$, and $f(3)=3$, then $f(2016)=$ ______.

(3) Toss a die and observe the number of points that come up, let event $A$ be the occurrence of an odd number, and event $B$ be the occurrence of $2$ points. Given $P(A)= \frac{1}{2},P(B)= \frac{1}{6}$, then the probability of getting an odd number or $2$ points is ______.

(4) $O$ is a fixed point on plane $\alpha$, $A$, $B$, $C$ are three vertices of $\triangle ABC$ on plane $\alpha$, $\angle B$, $\angle C$ are respectively the opposite angles of sides $AC$, $AB$. The correct statement(s) is(are) ______ (Write down all correct serial numbers).

① If moving point $P$ satisfies $\overrightarrow{OP}= \overrightarrow{OA}+ \overrightarrow{PB}+ \overrightarrow{PC}$, then the circumcenter of $\triangle ABC$ must be in the set of points $P$ that satisfy the condition;

② If moving point $P$ satisfies $\overrightarrow{OP}= \overrightarrow{OA}+\lambda\left( \frac{ \overrightarrow{AB}}{|AB|}+ \frac{ \overrightarrow{AC}}{|AC|}\right)(\lambda > 0)$, then the incenter of $\triangle ABC$ must be in the set of points $P$ that satisfy the condition;

③ If moving point $P$ satisfies $\overrightarrow{OP}= \overrightarrow{OA}+\lambda\left( \frac{ \overrightarrow{AB}}{|AB|\sin B}+ \frac{ \overrightarrow{AC}}{|AC|\sin C}\right)(\lambda > 0)$, then the centroid of $\triangle ABC$ must be in the set of points $P$ that satisfy the condition;

④ If moving point $P$ satisfies $\overrightarrow{OP}= \overrightarrow{OA}+\lambda\left( \frac{ \overrightarrow{AB}}{|AB|\cos B}+ \frac{ \overrightarrow{AC}}{|AC|\cos C}\right)(\lambda > 0)$, then the orthocenter of $\triangle ABC$ must be in the set of points $P$ that satisfy the condition;

⑤ If moving point $P$ satisfies $\overrightarrow{OP}= \frac{ \overrightarrow{OB}+ \overrightarrow{OC}}{2}+\lambda\left( \frac{ \overrightarrow{AB}}{|AB|\cos B}+ \frac{ \overrightarrow{AC}}{|AC|\cos C}\right)(\lambda > 0)$, then the circumcenter of $\triangle ABC$ must be in the set of points $P$ that satisfy the condition.

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