
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 $\triangle ABC$, the coordinates of points $A$ and $B$ are $A(5,-2)$ and $B(7,3)$, respectively. The midpoint $M$ of side $AC$ lies on the $y$-axis, and the midpoint $N$ of side $BC$ lies on the $x$-axis.

Find: $(1)$ the coordinates of point $C$; $(2)$ the equation of line $MN$; $(3)$ the area of the triangle formed by line $AB$ and the two coordinate axes.

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