
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 $W$ be a set composed of $n(n\geqslant 3)$ vectors in a plane. If $\overrightarrow{a} \in W$, and the magnitude of $\overrightarrow{a}$ is not less than the magnitude of the sum of all vectors in $W$ excluding $\overrightarrow{a}$ itself, then $\overrightarrow{a}$ is called a maximal vector of $W$. The following statements are given:  
$(1)$ If every vector in $W$ has the same direction, then there must exist a maximal vector in $W$;  
$(2)$ Given two non-collinear vectors $\overrightarrow{a}, \overrightarrow{b}$ in the plane, there always exists a unique plane vector $\overrightarrow{c} = -\overrightarrow{a} - \overrightarrow{b}$, such that every element in $W = \{\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}\}$ is a maximal vector;  
$(3)$ If every element in $W_{1} = \{\overrightarrow{a}_{1}, \overrightarrow{a}_{2}, \overrightarrow{a}_{3}\}, W_{2} = \{\overrightarrow{b}_{1}, \overrightarrow{b}_{2}, \overrightarrow{b}_{3}\}$ is a maximal vector, and $W_{1}, W_{2}$ have no common elements, then every element in $W_{1} \cup W_{2}$ is also a maximal vector.  
The correct statement(s) is/are \_\_\_\_\_\_.

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