
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
In triangle $ABC$, which is an equilateral triangle with side length $2$, given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $\overrightarrow{AB}=2\overrightarrow{a}$ and $\overrightarrow{AC}=2\overrightarrow{a}+\overrightarrow{b}$. Among the following conclusions, the correct ones are ______. (Write down all the correct conclusion numbers)

$①$ $\overrightarrow{a}$ is a unit vector; $②$ $\overrightarrow{b}$ is a unit vector; $③$ $\overrightarrow{a} \perp \overrightarrow{b}$; $④$ $\overrightarrow{b} \parallel \overrightarrow{BC}$; $⑤$ $(4\overrightarrow{a}+\overrightarrow{b}) \perp \overrightarrow{BC}$.

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