
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 $\triangle ABC$, it is known that $A(5, -2)$, $B(7, 3)$, and the midpoint $M$ of side $AC$ lies on the y-axis, while the midpoint $N$ of side $BC$ lies on the x-axis. Find:
(1) The coordinates of vertex $C$;
(2) The area of $\triangle ABC$.

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