
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 the Cartesian coordinate system $xOy$, for point $P(x,y)$, we call the point ${P'}(-y+1,x+1)$ the conjugate point of point $P$. Given that the conjugate point of point $A_{1}$ is $A_{2}$, the conjugate point of point $A_{2}$ is $A_{3}$, the conjugate point of point $A_{3}$ is $A_{4}$, and so on, obtaining points $A_{1}$, $A_{2}$, $A_{3}$, ..., $A_{n}$, and so on. If the coordinates of point $A_{1}$ are $(3,1)$, then the coordinates of point $A_{2}$ are ______, and the coordinates of point $A_{2023}$ are ______.

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