
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
Let $O$ be a fixed point on the plane $α$, and let $A$, $B$, and $C$ be three vertices of $\triangle ABC$ on the plane $α$, with $\angle B$ and $\angle C$ being the angles opposite to sides $AC$ and $AB$, respectively. Among the following statements, the correct one(s) is/are: (Write down all the correct serial numbers)
$①$ If the moving point $P$ satisfies $\overrightarrow{OP} = \overrightarrow{OA} + \overrightarrow{PB} + \overrightarrow{PC}$, then the circumcenter of $\triangle ABC$ is definitely in the set of points $P$ that satisfy the condition.
$②$ If the moving point $P$ satisfies $\overrightarrow{OP} = \overrightarrow{OA} + λ(\frac{\overrightarrow{AB}}{|AB|} + \frac{\overrightarrow{AC}}{|AC|}) (λ > 0)$, then the incenter of $\triangle ABC$ is definitely in the set of points $P$ that satisfy the condition.
$③$ If the moving point $P$ satisfies $\overrightarrow{OP} = \overrightarrow{OA} + λ(\frac{\overrightarrow{AB}}{|AB| \sin B} + \frac{\overrightarrow{AC}}{|AC| \sin C}) (λ > 0)$, then the centroid of $\triangle ABC$ is definitely in the set of points $P$ that satisfy the condition.
$④$ If the moving point $P$ satisfies $\overrightarrow{OP} = \overrightarrow{OA} + λ(\frac{\overrightarrow{AB}}{|AB| \cos B} + \frac{\overrightarrow{AC}}{|AC| \cos C}) (λ > 0)$, then the orthocenter of $\triangle ABC$ is definitely in the set of points $P$ that satisfy the condition.
$⑤$ If the moving point $P$ satisfies $\overrightarrow{OP} = \frac{\overrightarrow{OB} + \overrightarrow{OC}}{2} + λ(\frac{\overrightarrow{AB}}{|AB| \cos B} + \frac{\overrightarrow{AC}}{|AC| \cos C}) (λ > 0)$, then the circumcenter of $\triangle ABC$ is definitely in the set of points $P$ that satisfy the condition.

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