
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
For any point $P(x,y)$ in the plane rectangular coordinate system $xOy$, the following definition is given: <br/> Let $a=-x$, $b=x-y$. Then we call the points $M(a,b)$ and $N(b,a)$ as a pair of "beautiful points" of point $P$. For example, the pair of "beautiful points" of point $P(-1,2)$ are the points $(1,-3)$ and $(-3,1)$. <br/> $(1)$ The coordinates of a pair of "beautiful points" of point $A(4,1)$ are ______ and ______; <br/> $(2)$ If the pair of "beautiful points" of point $B(2,y)$ coincide, then the value of $y$ is ______; <br/> $(3)$ If one of the "beautiful points" of point $C$ has coordinates $(-2,7)$, find the coordinates of point $C$.

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