
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
The coordinates of the three vertices of triangle $\triangle ABC$ are $A(0,3)$, $B(-1,0)$, and $C(1,0)$. Xiao Hong translated $\triangle ABC$ to obtain $\triangle A'B'C'$ with the coordinates of the three vertices being $A'(0,0)$, $B'(-2,-3)$, and $C'(2,-3)$. <br/>$(1)$ Do you think the coordinates of the three vertices written by Xiao Hong are correct? Please explain your reasoning. <br/>$(2)$ If the y-coordinates of the three vertices written by Xiao Hong are all correct, and only one x-coordinate is correct, can you help Xiao Hong correctly write the coordinates of the three vertices?

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