
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
Define: In the Cartesian coordinate system, for point $P(x_{1}, y_{1})$, when point $Q(x_{2}, y_{2})$ satisfies $2(x_{1}+x_{2})=y_{1}+y_{2}$, point $Q(x_{2}, y_{2})$ is called a "doubling point" of point $P(x_{1}, y_{1})$. Given point $P_{1}(1,0)$, the correct conclusions are ______. (Fill in the numbers)  
① Points $Q_{1}(3,8)$ and $Q_{2}(-2,-2)$ are both "doubling points" of point $P_{1}$.  
② If point $A$ on the line $y=x+2$ is a "doubling point" of point $P_{1}$, then the coordinates of point $A$ are $\left(2,4\right)$.  
③ There exist two points on the parabola $y=x^{2}-2x-3$ that are "doubling points" of point $P_{1}$.  
④ If point $B$ is a "doubling point" of point $P_{1}$, then the minimum value of $P_{1}B$ is $\frac{4\sqrt{5}}{5}$.

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